ECR Cortiñas titelei 11.9.2008 11:10 Uhr Seite 1gnc/bibliographie/K-Theory/K-theory and... · An important connection between K-theory, topology, geometric group theory and noncommutative

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EMS Series of Congress Reports

EMS Series of Congress Reports publishes volumes originating from conferences or seminars focusingon any field of pure or applied mathematics. The individual volumes include an introduction into theirsubject and review of the contributions in this context. Articles are required to undergo a refereeing pro-cess and are accepted only if they contain a survey or significant results not published elsewhere in theliterature.

Previously published:Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowronski (ed.)


Guillermo CortiñasJoachim CuntzMax KaroubiRyszard NestCharles A. Weibel

Editors

K-Theory and Noncommutative Geometry


Editors:

Guillermo CortiñasDepartamento de MatemáticaFacultad de CienciasExactas y NaturalesUniversidad de Buenos AiresCiudad Universitaria Pab. 11428 Buenos Aires, ArgentinaEmail: [email protected]

2000 Mathematics Subject Classification: 19-06, 58-06; 14A22, 14Fxx, 46Lxx, 53D55, 58Bxx

Key words: Isomorphism conjectures, Waldhausen K-theory, twisted K-theory, equivariant cyclic homology, index theory, Bost–Connes algebra, C*-manifolds, T-duality, deformation of gerbes, torsiontheories, Parshin conjecture, Bloch–Kato conjecture

ISBN 978-3-03719-060-9

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© 2008 European Mathematical Society

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9 8 7 6 5 4 3 2 1

Joachim CuntzMathematisches InstitutWestfälische Wilhelms-UniversitätMünsterEinsteinstraße 6248149 Münster, GermanyEmail: [email protected]

Ryszard NestDepartment of MathematicsCopenhagen UniversityUniversitetsparken 52100 Copenhagen, DenmarkEmail: [email protected]

Charles A. WeibelDepartment of MathematicsRutgers UniversityHill Center – Busch Campus110 Frelinghuysen RoadPiscataway, NJ 08854-8019, U.S.A.Email: [email protected]

Max KaroubiUniversité Paris 72 place Jussieu75251 Paris, FranceEmail: [email protected]


Preface

This volume contains the proceedings ofVASBI, the ICM2006 satellite onK-theory andNoncommutative Geometry which took place in Valladolid, Spain, from August 31 toSeptember 6, 2006. The conference’s Scientific Committee was composed of JoachimCuntz, Max Karoubi, Ryszard Nest and Charles A. Weibel. The conference was madepossible through a grant by Caja Duero; additional funding was provided by the Uni-versity of Valladolid. Funding to cover expenses of US based participants was providedby NSF, through a grant to C. A. Weibel, who was responsible, first of all, for preparinga succesful funding application, and then for managing the funds. The local committee,composed of N. Abad and E. Ellis, were in charge of conference logistics.

I am indebted to all these people and institutions for their support.

Guillermo Cortiñas, Organizer

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Program list of speakers and topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Categorical aspects of bivariant K-theory

Ralf Meyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Inheritance of isomorphism conjectures under colimits

Arthur Bartels, Siegfried Echterhoff, and Wolfgang Lück . . . . . . . . . . . . . . . . . . . . . . . . 41

Coarse and equivariant co-assembly maps

Heath Emerson and Ralf Meyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

On K1 of a Waldhausen category

Fernando Muro and Andrew Tonks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Twisted K-theory – old and new

Max Karoubi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Equivariant cyclic homology for quantum groups

Christian Voigt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

A Schwartz type algebra for the tangent groupoid

Paulo Carrillo Rouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

C �-algebras associated with the ax C b-semigroup over N

Joachim Cuntz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

On a class of Hilbert C*-manifolds

Wend Werner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Duality for topological abelian group stacks and T -duality

Ulrich Bunke, Thomas Schick, Markus Spitzweck, and Andreas Thom . . . . . . . . . . . 227

viii Contents

Deformations of gerbes on smooth manifolds

Paul Bressler, Alexander Gorokhovsky, Ryszard Nest, and Boris Tsygan . . . . . . . . . 349

Torsion classes of finite type and spectra

Grigory Garkusha and Mike Prest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Parshin’s conjecture revisited

Thomas Geisser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Axioms for the norm residue isomorphism

Charles Weibel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

List of participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Introduction

Since its inception 50 years ago, K-theory has been a tool for understanding a wide-ranging family of mathematical structures and their invariants: topological spaces,rings, algebraic varieties and operator algebras are the dominant examples. The invari-ants range from characteristic classes in cohomology, through determinants of matrices,to Chow groups of varieties, as well as to traces and indices of elliptic operators. ThusK-theory is notable for its connections with other branches of mathematics.

Noncommutative geometry, on the other hand, develops tools which allow oneto think of noncommutative algebras in the same footing as commutative ones; asalgebras of functions on (noncommutative) spaces. The algebras in question come fromproblems in various areas of mathematics and mathematical physics; typical examplesinclude algebras of pseudodifferential operators, group algebras, and other algebrasarising from quantum field theory.

To study noncommutative geometric problems one considers invariants of the rel-evant noncommutative algebras. These invariants include algebraic and topologicalK-theory, and also cyclic homology, discovered independently by Alain Connes andBoris Tsygan, which can be regarded both as a noncommutative version of de Rhamcohomology, and as an additive version ofK-theory. There are primary and secondaryChern characters which pass from K-theory to cyclic homology. These characters arerelevant both to noncommutative and commutative problems, and have applicationsranging from index theorems to the detection of singularities of commutative algebraicvarieties.

The contributions to this volume represent this range of connections betweenK-theory, noncommmutative geometry, and other branches of mathematics.

An important connection betweenK-theory, topology, geometric group theory andnoncommutative geometry is given by the isomorphism conjectures, such as those dueto Baum–Connes, Bost–Connes and Farrell–Jones, which predict certain topologicalformulas for various kinds of K-theory of a crossed product, in both the C �-algebraicand the purely algebraic contexts. These problems have received a lot of attention overthe past twenty years. Not surprisingly, then, three of the articles in this volume discussthese problems. The first of these, by R. Meyer, is a survey on bivariant Kasparov theoryand E-theory. Both bivariant K-theories are approached via their universal propertiesand equipped with extra structure such as a tensor product and a triangulated categorystructure. The construction of the Baum–Connes assembly map via localisation ofcategories is reviewed and the relation with the purely topological construction byDavis and Lück is explained. In the second article, A. Bartels, S. Echterhoff andW. Lück investigate when Isomorphism Conjectures, such as the ones due to Baum–Connes, Bost and Farrell–Jones, are stable under colimits of groups over directed sets(with not necessarily injective structure maps). They show in particular that boththe K-theoretic Farrell–Jones Conjecture and the Bost Conjecture with coefficients

x Introduction

hold for those groups for which Higson, Lafforgue and Skandalis have disproved theBaum–Connes Conjecture with coefficients. H. Emerson and R. Meyer study in theircontribution an equivariant co-assembly map that is dual to the usual Baum–Connesassembly map and closely related to coarse geometry, equivariant Kasparov theory,and the existence of dual Dirac morphisms. As applications, they prove the existenceof dual Dirac morphisms for groups with suitable compactifications, that is, satisfyingthe Carlsson–Pedersen condition, and study a K-theoretic counterpart to the properLipschitz cohomology of Connes, Gromov and Moscovici.

Many applications of K-theory to geometric topology come from the K-theory ofWaldhausen categories. The article by F. Muro and A. Tonks is about theK-theory of aWaldhausen category. They give a simple representation of all elements in K1 of sucha category and prove relations between these representatives which hold in K1.

Another connection between K-theory, topology and analysis comes from twistedK-theory, which has received renewed attention in the last decade, in the light of newdevelopments inspired by Mathematical Physics. The next article is a survey on thissubject, written by M. Karoubi, one of its founders. The author also proves some newresults in the subject: a Thom isomorphism, explicit computations in the equivariantcase and new cohomology operations.

As mentioned above, one of the most interesting points of contact betweenK-theoryand noncommutative geometry comes through cyclic homology. C.Voigt’s contributionis about this theory. He defines equivariant periodic cyclic homology for bornologicalquantum groups. Generalizing corresponding results from the group case, he shows thatthe theory is homotopy invariant, stable, and satisfies excision in both variables. Alongthe way he proves Radford’s formula for the antipode of a bornological quantum group.Moreover he discusses anti-Yetter–Drinfeld modules and establishes an analogue of theTakesaki–Takai duality theorem in the setting of bornological quantum groups.

A central theme in noncommutative geometry is index theory. A basic constructionin this area consists of associating a C �-algebra to a geometric problem, and thencompute an index usingK-theory. P. Carrillo Rouse’s article deals with the problem ofindex theory on singular spaces, and in particular with the construction of algebras andindices between the enveloping C �-algebra and the convolution algebra of compactlysupported functions.

The article by J. Cuntz is also about C �-algebras. It presents a C �-algebra whichis naturally associated to the axC b-semigroup over N. It is simple and purely infiniteand can be obtained from the algebra considered by Bost and Connes by adding oneunitary generator which corresponds to addition. Its stabilization can be described asa crossed product of the algebra of continuous functions, vanishing at infinity, on thespace of finite adeles for Q by the natural action of the ax C b-group over Q.

W. Werner applies C �-algebraic methods in infinite dimensional differential geom-etry in his contribution. It deals with a special class of ‘non-compact’ hermitian infinitedimensional symmetric spaces, generically denoted by U . The author calculates theirinvariant connection very explicitly and uses the concept of a Hilbert C*-manifold sothat the Banach manifold in question is of the form AutU=H , where AutU is theautomorphism group of the Hilbert C*-manifold. Using results previously obtained

Introduction xi

with D. Blecher, he characterizes causal structure on U that comes from interpretingthe elements of U as bounded Hilbert space operators.

The subject of the next article, by U. Bunke, T. Schick, M. Spitzweck and A. Thom,connected with group theory, topology, as well as to mathematical physics and non-commutative geometry, is Pontrjagin duality. The authors extend Pontrjagin dualityfrom topological abelian groups to certain locally compact group stacks. To this endthey develop a sheaf theory on the big site of topological spaces S in order to prove thatthe sheaves ExtiShAbS.G;T /, i D 1; 2, vanish, where G is the sheaf represented by alocally compact abelian group and T is the circle. As an application of the theory theyinterpret topological T -duality of principal Tn-bundles in terms of Pontrjagin dualityof abelian group stacks.

An important source of examples and problems in noncommutative geometry comesfrom deformations of commutative structures. In this spirit, the article by P. Bressler,A. Gorokhovsky, R. Nest and B. Tsygan investigates the 2-groupoid of deformationsof a gerbe on a C1 manifold. They identify the latter with the Deligne 2-groupoid of acorresponding twist of the differential graded Lie algebra of local Hochschild cochainson C1 functions.

The next article, by G. Garkusha and M. Prest, provides a bridge between thenoncommutative and the commutative worlds. A common approach in noncommutativegeometry is to study abelian or triangulated categories and to think of them as thereplacement of an underlying scheme. This idea goes back to work of Grothendieckand Gabriel and continues to be of interest. The approach is justified by the fact thata (commutative noetherian) scheme can be reconstructed from the abelian category ofquasi-coherent sheaves (Gabriel) or from the category of perfect complexes (Balmer). Anatural question to ask, is whether the hypothesis on the scheme to be noetherian is reallynecessary. This is precisely the theme of recent work of Garkusha and Prest. They showhow to deal with affine schemes, without imposing any finiteness assumptions. Morespecifically, the article in this volume discusses for module categories over commutativerings the classification of torsion classes of finite type. For instance, the prime idealspectrum can be reconstructed from this classification.

The final two contributions are aboutK-theory and (commutative) algebraic geom-etry.

T. Geisser’s article is about Parshin’s conjecture. The latter states thatKi .X/Q D 0for i > 0 and X smooth and projective over a finite field Fq . The purpose ofthis article is to break up Parshin’s conjecture into several independent statements,in the hope that each of them is easier to attack individually. If CHn.X; i/ is Bloch’shigher Chow group of cycles of relative dimension n, then in view of Ki .X/Q ŠLn CHn.X; i/Q, Parshin’s conjecture is equivalent to Conjecture P.n/ for all n, stat-

ing that CHn.X; i/Q D 0 for i > 0, and all smooth and projective X . Assumingresolution of singularities, the author shows that Conjecture P.n/ is equivalent to theconjunction of three conjectures A.n/, B.n/ and C.n/, and gives several equivalentversions of these conjectures.

Finally, C. Weibel’s article gives an axiomatic framework for proving that the normresidue map is an isomorphism (i.e., for settling the motivic Bloch–Kato conjecture).

xii Introduction

This framework is a part of the Voevodsky–Rost program.

In conclusion, this volume presents a good sample of the wide range of aspects ofcurrent research in K-theory, noncommutative geometry and their applications.

Guillermo Cortiñas, Joachim Cuntz, Max Karoubi,Ryszard Nest, and Charles A. Weibel

Program list of speakers and topics

There were four courses – each consisting of three lectures – nine invited talks, andtwelve short communications.

Courses

Joachim Cuntz Cyclic homology

Bernhard Keller DG-categories

Boris Tsygan Characteristic classes in noncommutative geometry

Charles Weibel Motivic cohomology

Invited talks

Paul Balmer Triangular Geometry

Stefan Gille A Gersten–Witt complex for Azumaya algebras with involution

Victor Ginzburg Calabi–Yau algebras

Alexander Gorokhovsky Deformation quantization of étale groupoids and gerbes

Maxim Kontsevich On the degeneration of the Hodge to de Rham spectralsequence

Marc Levine Algebraic cobordism

Wolfgang Lück The Farrell–Jones Conjecture for algebraic K-theory holdsfor word-hyperbolic groups and arbitrary coefficients

Ralf Meyer Bivariant K-theory: where topology and analysis meet

Amnon Neeman The homotopy category of flat modules and Grothendieck’slocal duality

Ryszard Nest Applications of triangulated structure of KK-theory to quantumgroups

Andreas Thom Comparison between algebraic and topological K-theory

Mariusz Wodzicki Megatrace

Communications

Marian Anton Homological symbols

Grzegorz Banaszak An imbedding property of K-groups

Ulrich Bunke Duality of topological abelian groups stacks and T -duality

Leandro Cagliero The cohomology ring of truncated quiver algebras

Paulo Carrillo Rouse An analytical index for Lie groupoids

xiv Program list of speakers and topics

Guillermo Cortiñas K-theory and homological obstructions to regularity

Mathias Fuchs Cyclic cohomology associated to higher rank latticesand the idempotents of the group ring

Immaculada Gálvez Up-to-homotopy structures on vertex algebras

Grigory Garkusha Triangulated categories of rings

Juan José Guccione Relative cyclic homology of square zero extensions

Fernando Muro On the 1-type of Waldhausen K-theory

Wend Werner Ternary rings of operators, affine manifolds and causal structure

Categorical aspects of bivariant K-theory

Ralf Meyer

1 Introduction

Non-commutative topology deals with topological properties of C�-algebras. Alreadyin the 1970s, the classification of AF-algebras by K-theoretic data [15] and the workof Brown–Douglas–Fillmore on essentially normal operators [6] showed clearly thattopology provides useful tools to study C�-algebras. A breakthrough was Kasparov’sconstruction of a bivariant K-theory for separable C�-algebras. Besides its applicationswithin C�-algebra theory, it also yields results in classical topology that are hard or evenimpossible to prove without it. A typical example is the Novikov conjecture, whichdeals with the homotopy invariance of certain invariants of smooth manifolds with agiven fundamental group. This conjecture has been verified for many groups usingKasparov theory, starting with [31]. The C�-algebraic formulation of the Novikovconjecture is closely related to the Baum–Connes conjecture, which deals with thecomputation of the K-theory K�.C�redG/ of reduced group C�-algebras and has beenone of the centres of attention in non-commutative topology in recent years.

The Baum–Connes conjecture in its original formulation [4] only deals with a singleK-theory group; but a better understanding requires a different point of view. Theapproach by Davis and Lück in [14] views it as a natural transformation between twohomology theories for G-CW-complexes. An analogous approach in the C�-algebraframework appeared in [38]. These approaches to the Baum–Connes conjecture showthe importance of studying not just single C�-algebras, but categories of C�-algebrasand their properties. Older ideas like the universal property of Kasparov theory areof the same nature. Studying categories of objects instead of individual objects isbecoming more and more important in algebraic topology and algebraic geometry aswell.

Several mathematicians have suggested, therefore, to apply general constructionswith categories (with additional structure) like generators, Witt groups, the centre, andsupport varieties to the C�-algebra context. Despite the warning below, this seems apromising project, where little has been done so far. To prepare for this enquiry, wesummarise some of the known properties of categories of C�-algebras; we cover ten-sor products, some homotopy theory, universal properties, and triangulated structures.In addition, we examine the Universal Coefficient Theorem and the Baum–Connesassembly map.

Despite many formal similarities, the homotopy theory of spaces and non-commu-tative topology have a very different focus.

On the one hand, most of the complexities of the stable homotopy category of spacesvanish for C�-algebras because only very few homology theories for spaces have anon-commutative counterpart: any functor on C�-algebras satisfying some reasonable

2 R. Meyer

assumptions must be closely related to K-theory. Thus special features of topologicalK-theory become more transparent when we work with C�-algebras.

On the other hand, analysis may create new difficulties, which appear to be very hardto study topologically. For instance, there exist C�-algebras with vanishing K-theorywhich are nevertheless non-trivial in Kasparov theory; this means that the UniversalCoefficient Theorem fails for them. I know no non-trivial topological statement aboutthe subcategory of the Kasparov category consisting of C�-algebras with vanishingK-theory; for instance, I know no compact objects.

It may be necessary, therefore, to restrict attention to suitable “bootstrap” categoriesin order to exclude pathologies that have nothing to do with classical topology. Moreor less by design, the resulting categories will be localisations of purely topologicalcategories, which we can also construct without mentioning C�-algebras. For instance,we know that the Rosenberg–Schochet bootstrap category is equivalent to a full sub-category of the category of BU -module spectra. But we can hope for more interestingcategories when we work equivariantly with respect to, say, discrete groups.

2 Additional structure in C�-algebra categories

We assume that the reader is familiar with some basic properties of C�-algebras, includ-ing the definition (see for instance [2], [13]). As usual, we allow non-unital C�-algebras.We define some categories of C�-algebras in §2.1 and consider group C�-algebras andcrossed products in §2.2. Then we discuss C�-tensor products and mention the notionsof nuclearity and exactness in §2.3. The upshot is that C�alg and G-C�alg carry twostructures of symmetric monoidal category, which coincide for nuclear C�-algebras.We prove in §2.4 that C�alg andG-C�alg are bicomplete, that is, all diagrams in themhave both a limit and a colimit. We equip morphism spaces between C�-algebras with acanonical base point and topology in §2.5; thus the category of C�-algebras is enrichedover the category of pointed topological spaces. In §2.6, we define mapping cones andcylinders in categories of C�-algebras; these rudimentary tools suffice to carry oversome basic homotopy theory.

2.1 Categories of C�-algebras

Definition 1. The category of C�-algebras is the category C�alg whose objects arethe C�-algebras and whose morphismsA! B are the �-homomorphismsA! B; wedenote this set of morphisms by Hom.A;B/.

A C�-algebra is called separable if it has a countable dense subset. We often restrictattention to the full subcategory C�sep � C�alg of separable C�-algebras.

Examples of C�-algebras are group C�-algebras and C�-crossed products. Webriefly recall some relevant properties of these constructions. A more detailed discus-sion can be found in many textbooks such as [44].

Categorical aspects of bivariant K-theory 3

Definition 2. We write A 22 C to denote that A is an object of the category C. Thenotation f 2 C means that f is a morphism in C; but to avoid confusion we alwaysspecify domain and target and write f 2 C.A;B/ instead of f 2 C.

2.2 Group actions, and crossed products. For any locally compact group G, wehave a reduced group C�-algebra C�red.G/ and a full group C�-algebra C�.G/. Both aredefined as completions of the group Banach algebra .L1.G/;�/ for suitable C�-norms.They are related by a canonical surjective �-homomorphism C�.G/! C�red.G/, whichis an isomorphism if and only if G is amenable.

The norm on C�.G/ is the maximal C�-norm, so that any strongly continuousunitary representation of G on a Hilbert space induces a �-representation of C�.G/.The norm on C�red.G/ is defined using the regular representation ofG onL2.G/; hence arepresentation ofG only induces a �-representation of C�red.G/ if it is weakly containedin the regular representation. For reductive Lie groups and reductive p-adic groups,these representations are exactly the tempered representations, which are much easierto classify than all unitary representations.

Definition 3. A G-C�-algebra is a C�-algebra A with a strongly continuous represen-tation of G by C�-algebra automorphisms. The category of G-C�-algebras is the cat-egory G-C�alg whose objects are the G-C�-algebras and whose morphisms A ! B

are the G-equivariant �-homomorphisms A ! B; we denote this morphism set byHomG.A;B/.

Example 4. If G D Z, then a G-C�-algebra is nothing but a pair .A; ˛/ consisting ofa C�-algebra A and a �-automorphism ˛ W A! A: let ˛ be the action of the generator1 2 Z.

Equipping C�-algebras with a trivial action provides a functor

� W C�alg! G-C�alg; A 7! A� : (1)

Since C has only the identity automorphism, the trivial action is the only way to turn Cinto a G-C�-algebra.

The full and reduced C�-crossed products are versions of the full and reduced groupC�-algebras with coefficients in G-C�-algebras (see [44]). They define functors

G Ë ; G Ër W G-C�alg! C�alg; A 7! G Ë A; G Ër A;

such that G Ë C D C�.G/ and G Ër C D C�red.G/.

Definition 5. A diagram I ! E ! Q in C�alg is an extension if it is isomorphic tothe canonical diagram I ! A! A=I for some ideal I in a C�-algebra A; extensionsinG-C�alg are defined similarly, usingG-invariant ideals inG-C�-algebras. We writeI � E � Q to denote extensions.

Although C�-algebra extensions have some things in common with extensions of,say, modules, there are significant differences because C�alg is not Abelian, not evenadditive.

4 R. Meyer

Proposition 6. The full crossed product functor G Ë W G-C�alg ! C�alg is exactin the sense that it maps extensions in G-C�alg to extensions in C�alg.

Proof. This is Lemma 4.10 in [21].

Definition 7. A locally compact groupG is called exact if the reduced crossed productfunctor G Ër W G-C�alg! C�alg is exact.

Although this is not apparent from the above definition, exactness is a geometricproperty of a group: it is equivalent to Yu’s property (A) or to the existence of anamenable action on a compact space [43].

Most groups you know are exact. The only source of non-exact groups known at themoment are Gromov’s random groups. Although exactness might remind you of thenotion of flatness in homological algebra, it has a very different flavour. The differenceis that the functor G Ër always preserves injections and surjections. What may gowrong for non-exact groups is exactness in the middle (compare the discussion beforeProposition 18). Hence we cannot study the lack of exactness by derived functors.

Even for non-exact groups, there is a class of extensions for which reduced crossedproducts are always exact:

Definition 8. A section for an extension

Ii� E

p� Q (2)

inG-C�alg is a map (of sets) s W Q! E withpıs D idQ. We call (2) split if there is asection that is a G-equivariant �-homomorphism. We call (2) G-equivariantly cp-splitif there is a G-equivariant, completely positive, contractive, linear section.

Sections are also often called lifts, liftings, or splittings.

Proposition 9. Both the reduced and the full crossed product functors map split ex-tensions in G-C�alg again to split extensions in C�alg and G-equivariantly cp-splitextensions in G-C�alg to cp-split extensions in C�alg.

Proof. Let Ki� E

p� Q be an extension in G-C�alg. Proposition 6 shows that

G Ë K � G Ë E � G Ë Q is again an extension. Since reduced and full crossedproducts are functorial for equivariant completely positive contractions, this extensionis split or cp-split if the original extension is split or equivariantly cp-split, respectively.This yields the assertions for full crossed products.

Since a �-homomorphism with dense range is automatically surjective, the inducedmap G Ër p W G Ër E ! G Ër Q is surjective. It is evident from the definition ofreduced crossed products that G Ër i is injective. What is unclear is whether therange of G Ër i and the kernel of G Ër p coincide. As for the full crossed product, aG-equivariant completely positive contractive section s W Q! E induces a completelypositive contractive section G Ër s for G Ër p. The linear map

' WD idGËrE � .G Ër s/ ı .G Ër p/ W G Ër E ! G Ër E


is a retraction from G Ër E onto the kernel of G Ër p by construction. Furthermore, itmaps the dense subspace L1.G;E/ into L1.G;K/. Hence it maps all of G Ër E intoG Ër K. This implies G Ër K D ker.G Ër p/ as desired.

2.3 Tensor products and nuclearity. Most results in this section are proved in detailin [42], [60]. Let A1 and A2 be two C�-algebras. Their (algebraic) tensor productA1 ˝ A2 is still a �-algebra. A C�-tensor product of A1 and A2 is a C�-completionof A1 ˝ A2, that is, a C�-algebra that contains A1 ˝ A2 as a dense �-subalgebra. AC�-tensor product is determined uniquely by the restriction of its norm to A1˝A2. Anorm on A1 ˝ A2 is allowed if it is a C�-norm, that is, multiplication and involutionhave norm 1 and kx�xk D kxk2 for all x 2 A1 ˝ A2.

There is a maximal C�-norm on A1 ˝ A2. The resulting C�-tensor product iscalled maximal C�-tensor product and denoted A1 ˝max A2. It is characterised by thefollowing universal property:

Proposition10. There is anatural bijectionbetweennon-degenerate �-homomorphismsA1 ˝max A2 ! B.H / and pairs of commuting non-degenerate �-homomorphismsA1 ! B.H / and A2 ! B.H /; here we may replace B.H / by any multiplier algebraM.D/ of a C�-algebraD.

A �-representation A! B.H / is non-degenerate if A �H is dense in H ; we needthis to get representations ofA1 andA2 out of a representation ofA1˝maxA2 because,for non-unital algebras, A1 ˝max A2 need not contain copies of A1 and A2.

The maximal tensor product is natural, that is, it defines a bifunctor

˝max W C�alg � C�alg! C�alg:

If A1 and A2 are G-C�-algebras, then A1 ˝max A2 inherits two group actions of G bynaturality; these are again strongly continuous, so that A1˝max A2 becomes a G �G-C�-algebra. Restricting the action to the diagonal in G �G, we turn A1 ˝max A2 intoa G-C�-algebra. Thus we get a bifunctor

˝max W G-C�alg �G-C�alg! G-C�alg:

The following lemma asserts, roughly speaking, that this tensor product has thesame formal properties as the usual tensor product for vector spaces:

Lemma 11. There are canonical isomorphisms

.A˝max B/˝max C Š A˝max .B ˝max C/;

A˝max B Š B ˝max A;

C ˝max A Š A Š A˝max C

for all objects of G-C�alg (and, in particular, of C�alg). These define a structure ofsymmetric monoidal category on G-C�alg (see [35], [52]).

6 R. Meyer

A functor between symmetric monoidal categories is called symmetric monoidal ifit is compatible with the tensor products in a suitable sense [52]. A trivial example isthe functor � W C�alg! G-C�alg that equips a C�-algebra with the trivial G-action.

It follows from the universal property that ˝max is compatible with full crossedproducts: if A 22 G-C�alg, B 22 C�alg, then there is a natural isomorphism

G Ë�A˝max �.B/

� Š .G Ë A/˝max B: (3)

Like full crossed products, the maximal tensor product may be hard to describe becauseit involves a maximum of all possible C�-tensor norms. There is another C�-tensor normthat is defined more concretely and that combines well with reduced crossed products.

Recall that any C�-algebra A can be represented faithfully on a Hilbert space. Thatis, there is an injective �-homomorphism A! B.H / for some Hilbert space H ; hereB.H / denotes the C�-algebra of bounded operators on H . IfA is separable, we can findsuch a representation on the separable Hilbert space H D `2.N/. The tensor product oftwo Hilbert spaces H1 and H2 carries a canonical inner product and can be completedto a Hilbert space, which we denote by H1 x H2. If H1 and H2 support faithfulrepresentations of C�-algebras A1 and A2, then we get an induced �-representation ofA1 ˝ A2 on H1 x H2.

Definition 12. The minimal tensor product A1˝min A2 is the completion of A1˝A2with respect to the operator norm from B.H1 x H2/.

It can be check that this is well-defined, that is, the C�-norm on A1 ˝ A2 does notdepend on the chosen faithful representations of A1 and A2. The same argument alsoyields the naturality of A1 ˝min A2. Hence we get a bifunctor

˝min W G-C�alg �G-C�alg! G-C�algIit defines another symmetric monoidal category structure on G-C�alg.

We may also call A1 ˝min A2 the spatial tensor product. It is minimal in the sensethat it is dominated by any C�-tensor norm onA1˝A2 that is compatible with the givennorms on A1 and A2. In particular, we have a canonical surjective �-homomorphism

A1 ˝max A2 ! A1 ˝min A2: (4)

Definition 13. A C�-algebra A1 is nuclear if the map in (4) is an isomorphism for allC�-algebras A2.

The name comes from an analogy between nuclear C�-algebras and nuclear locallyconvex topological vector spaces (see [20]). But this is merely an analogy: the onlyC�-algebras that are nuclear as locally convex topological vector spaces are the finite-dimensional ones.

Many important C�-algebras are nuclear. This includes the following examples:

• commutative C�-algebras;

• matrix algebras and algebras of compact operators on Hilbert spaces;


• group C�-algebras of amenable groups (or groupoids);

• C�-algebras of type I and, in particular, continuous trace C�-algebras.

If A is nuclear, then there is only one reasonable C�-algebra completion of A˝B .Therefore, if we can write down any, it must be equal to both A˝min B and A˝max B .

Example 14. For a compact space X and a C�-algebra A, we let C.X;A/ be theC�-algebra of all continuous functions X ! A. If X is a pointed compact space, welet C0.X;A/ be the C�-algebra of all continuous functions X ! A that vanish at thebase point ofX ; this contains C.X;A/ as a special case because C.X;A/ Š C0.XC; A/,where XC D X t f?g with base point ?. We have

C0.X;A/ Š C0.X/˝min A Š C0.X/˝max A:

Example 15. There is a unique C�-norm on Mn ˝ A DMn.A/ for all n 2 N.For a Hilbert space H , let K.H / be the C�-algebra of compact operators on H .

Then K.H /˝A contains copies of Mn.A/, n 2 N, for all finite-dimensional subspacesof H . These carry a unique C�-norm. The C�-norms on these subspaces are compatibleand extend to the unique C�-norm on K.H /˝ A.

The class of nuclear C�-algebras is closed under ideals, quotients (by ideals), exten-sions, inductive limits, and crossed products by actions of amenable locally compactgroups. In particular, this covers crossed products by automorphisms (see Example 4).

C�-subalgebras of nuclear C�-algebras need not be nuclear any more, but they stillenjoy a weaker property called exactness:

Definition 16. A C�-algebra A is called exact if the functor A ˝min preservesC�-algebra extensions.

It is known [33], [43] that a discrete group is exact (Definition 7) if and only if itsgroup C�-algebra is exact (Definition 16), if and only if the group has an amenableaction on some compact topological space.

Example 17. LetG be the non-Abelian free group on 2 generators. LetG act freely andproperly on a tree as usual. Let X be the ends compactification of this tree, equippedwith the induced action of G. This action is known to be amenable, so that G is anexact group. Since the action is amenable, the crossed product algebrasGËr C.X/ andGËC.X/ coincide and are nuclear. The embedding C ! C.X/ induces an embeddingC�red.G/! G Ë C.X/. ButG is not amenable. Hence the C�-algebra C�red.G/ is exactbut not nuclear.

As for crossed products, ˝min respects injections and surjections. The issue withexactness in the middle is the following. Elements of A ˝min B are limits of tensorsof the form

PniD1 ai ˝ bi with a1; : : : ; an 2 A, b1; : : : ; bn 2 B . If an element

in A ˝min B is annihilated by the map to .A=I / ˝min B , then we can approximateit by such finite sums for which

PniD1.ai mod I / ˝ bi goes to 0. But this does

not suffice to find approximations in I ˝ B . Thus the kernel of the projection mapA˝min B � .A=I /˝min B may be strictly larger than I ˝min B .

8 R. Meyer

Proposition 18. The functor ˝max is exact in each variable, that is, ˝max D mapsextensions in G-C�alg again to extensions for eachD 22 G-C�alg.

Both ˝min and ˝max map split extensions to split extensions and (equivariantly)cp-split extensions again to (equivariantly) cp-split extensions.

The proof is similar to the proofs of Propositions 6 and 9.If A or B is nuclear, we simply write A˝ B for A˝max B Š A˝min B .

2.4 Limits and colimits

Proposition 19. The categories C�alg and G-C�alg are bicomplete, that is, any(small) diagram in these categories has both a limit and a colimit.

Proof. To get general limits and colimits, it suffices to construct equalisers and co-equalisers for pairs of parallel morphisms f0; f1 W A � B , direct products and coprod-uctsA1�A2 andA1tA2 for any pair of objects and, more generally, for arbitrary setsof objects.

The equaliser and coequaliser of f0; f1 W A � B are

ker.f0 � f1/ D fa 2 A j f0.a/ D f1.a/g � Aand the quotient of A1 by the closed �-ideal generated by the range of f0 � f1, re-spectively. Here we use that quotients of C�-algebras by closed �-ideals are againC�-algebras. Notice that ker.f0 � f1/ is indeed a C�-subalgebra of A.

The direct productA1�A2 is the usual direct product, equipped with the canonicalC�-algebra structure. We can generalise the construction of the direct product to infinitedirect products: let

Qi2I Ai be the set of all norm-bounded sequences .ai /i2I with

ai 2 Ai for all i 2 I ; this is a C�-algebra with respect to the obvious �-algebra structureand the norm k.ai /k WD supi2Ikaik. It has the right universal property because any�-homomorphism is norm-contracting. (A similar construction with Banach algebraswould fail at this point.)

The coproductA1tA2 is also called free product and denotedA1�A2; its construc-tion is more involved. The free C-algebra generated by A1 and A2 carries a canonicalinvolution, so that it makes sense to study C�-norms on it. It turns out that there is amaximal such C�-norm. The resulting C�-completion is the free product C�-algebra.In the equivariant case, A1 tA2 inherits an action of G, which is strongly continuous.The resulting object of G-C�alg has the correct universal property for a coproduct.

An inductive system of C�-algebras .Ai ; ˛ji /i2I is called reduced if all the maps

˛ji W Ai ! Aj are injective; then they are automatically isometric embeddings. We

may as well assume that these maps are identical inclusions of C�-subalgebras. Thenwe can form a �-algebra

SAi , and the given C�-norms piece together to a C�-norm

onSAi . The resulting completion is lim�!.Ai ; ˛

ji /. In particular, we can construct an

infinite coproduct as the inductive limit of its finite sub-coproducts. Thus we get infinitecoproducts.


The category of commutative C�-algebras is equivalent to the opposite of the cat-egory of pointed compact spaces by the Gelfand–Naimark Theorem. It is frequentlyconvenient to replace a pointed compact space X with base point ? by the locallycompact space X n f?g. A continuous map X ! Y extends to a pointed continuousmap XC ! YC if and only if it is proper. But there are more pointed continuousmaps f W XC ! YC than proper continuous maps X ! Y because points in X maybe mapped to the point at infinity 1 2 YC. For instance, the zero homomorphismC0.Y /! C0.X/ corresponds to the constant map x 7! 1.

Example 20. If U � X is an open subset of a locally compact space, then C0.U / isan ideal in C0.X/. No map X ! U corresponds to the embedding C0.U /! C0.X/.

Example 21. Products of commutative C�-algebras are again commutative and cor-respond by the Gelfand–Naimark Theorem to coproducts in the category of pointedcompact spaces. The coproduct of a set of pointed compact spaces is the Stone–Cechcompactification of their wedge sum. Thus infinite products in C�alg and G-C�algdo not behave well for the purposes of homotopy theory.

The coproduct of two non-zero C�-algebras is never commutative and hence has noanalogue for (pointed) compact spaces. The smash product for pointed compact spacescorresponds to the tensor product of C�-algebras because

C0.X ^ Y / Š C0.X/˝min C0.Y /:

2.5 Enrichment over pointed topological spaces. LetA andB be C�-algebras. It iswell-known that a �-homomorphism f W A! B is automatically norm-contracting andinduces an isometric embedding A= ker f ! B with respect to the quotient norm onA= ker f . The reason for this is that the norm for self-adjoint elements in a C�-algebraagrees with the spectral radius and hence is determined by the algebraic structure; bythe C�-condition kak2 D ka�ak, this extends to all elements of a C�-algebra.

It follows that Hom.A;B/ is an equicontinuous set of linear maps A! B . We al-ways equip Hom.A;B/with the topology of pointwise norm-convergence. Its subbasicopen subsets are of the form

ff W A! B j k.f � f0/.a/k < 1 for all a 2 Sgfor f0 2 Hom.A;B/ and a finite subset S � A. Since Hom.A;B/ is equicontinuous,this topology agrees with the topology of uniform convergence on compact subsets,which is generated by the corresponding subsets for compact S . This is nothing but thecompact-open topology on mapping spaces. But it differs from the topology definedby the operator norm. We shall never use the latter.

Lemma 22. If A is separable, then Hom.A;B/ is metrisable for any B .

Proof. There exists a sequence .an/n2N inAwith lim an D 0whose closed linear spanis all of A. The metric

d.f1; f2/ D supfkf1.an/ � f2.an/k j n 2 Ng

10 R. Meyer

defines the topology of uniform convergence on compact subsets on Hom.A;B/ be-cause the latter is equicontinuous.

There is a distinguished element in Hom.A;B/ as well, namely, the zero homomor-phism A! 0! B . Thus Hom.A;B/ becomes a pointed topological space.

Proposition 23. The above construction provides an enrichment of C�alg over thecategory of pointed Hausdorff topological spaces.

Proof. It is clear that 0 ı f D 0 and f ı 0 D 0 for all morphisms f . Furthermore, wemust check that composition of morphisms is jointly continuous. This follows fromthe equicontinuity of Hom.A;B/.

This enrichment allows us to carry over some important definitions from cate-gories of spaces to C�alg. For instance, a homotopy between two �-homomorphismsf0; f1 W A ! B is a continuous path between f0 and f1 in the topological spaceHom.A;B/. In the following proposition, MapC.X; Y / denotes the space of mor-phisms in the category of pointed topological spaces, equipped with the compact-opentopology.

Proposition 24 (compare Proposition 3.4 in [28]). Let A and B be C�-algebras andlet X be a pointed compact space. Then

MapC�X;Hom.A;B/

� Š Hom�A;C0.X;B/

�as pointed topological spaces.

Proof. If we view A and B as pointed topological spaces with the norm topology andbase point 0, then Hom.A;B/ � MapC.A;B/ is a topological subspace with the samebase point because Hom.A;B/ also carries the compact-open topology. Since X iscompact, standard point set topology yields homeomorphisms

MapC�X;MapC.A;B/

� Š MapC.X ^ A;B/Š MapC

�A;MapC.X;B/

� D MapC�A;C0.X;B/

�:

These restrict to the desired homeomorphism.

In particular, a homotopy between two �-homomorphisms f0; f1 W A � B is equiv-alent to a �-homomorphism f W A ! C.Œ0; 1�; B/ with evt ı f D ft for t D 0; 1,where evt denotes the �-homomorphism

evt W C.Œ0; 1�; B/! B; f 7! f .t/:

We also haveC0

�X;C0.Y; A/

� Š C0.X ^ Y;A/for all pointed compact spaces X , Y and all G-C�-algebras A. Thus a homotopybetween two homotopies can be encoded by a �-homomorphism

A! C�Œ0; 1�;C.Œ0; 1�; B/

� Š C.Œ0; 1�2; B/:


These constructions work only for pointed compact spaces. If we enlarge thecategory of C�-algebras to a suitable category of projective limits of C�-algebras asin [28], then we can define C0.X;A/ for any pointed compactly generated space X .But we lose some of the nice analytic properties of C�-algebras. Therefore, I prefer tostick to the category of C�-algebras itself.

2.6 Cylinders, cones, and suspensions. The following definitions go back to [54],where some more results can be found. The description of homotopies above leads usto define the cylinder over a C�-algebra A by

Cyl.A/ WD C.Œ0; 1�; A/:

This is compatible with the cylinder construction for spaces because

Cyl�C0.X/

� Š C�Œ0; 1�;C0.X/

� Š C0.Œ0; 1�C ^X/for any pointed compact space X ; if we use locally compact spaces, we get Œ0; 1� �Xinstead of Œ0; 1�C ^X .

The universal property of Cyl.A/ is dual to the usual one for spaces because theidentification between pointed compact spaces and commutative C�-algebras is con-travariant.

Similarly, we may define the cone Cone.A/ and the suspension Sus.A/ by

Cone.A/ WD C0�Œ0; 1� n f0g; A/; Sus.A/ WD C0

�Œ0; 1� n f0; 1g; A/ Š C0.S

1; A/;

where S1 denotes the pointed 1-sphere, that is, circle. These constructions are compat-ible with the corresponding ones for spaces as well, that is,

Cone�C0.X/

� Š C0.Œ0; 1� ^X/; Sus�C0.X/

� Š C0.S1 ^X/:

Here Œ0; 1� has the base point 0.

Definition 25. Let f W A ! B be a morphism in C�alg or G-C�alg. The mappingcylinder Cyl.f / and the mapping cone Cone.f / of f are the limits of the diagrams

Af�! B

ev1 �� Cyl.B/; Af�! B

ev1 �� Cone.B/:

More concretely,

Cone.f / D ˚.a; b/ 2 A � C0

�.0; 1�; B

� j f .a/ D b.1/�;Cyl.f / D ˚

.a; b/ 2 A � C0.Œ0; 1�; B� j f .a/ D b.1/�:

If f W X ! Y is a morphism of pointed compact spaces, then the mapping coneand mapping cylinder of the induced �-homomorphism C0.f / W C0.Y /! C0.X/ agreewith C0

�Cyl.f /

�and C0

�Cone.f /

�, respectively.

The cylinder, cone, and suspension functors are exact for various kinds of extensions:they map extensions, split extensions, and cp-split extensions again to extensions, split

12 R. Meyer

extensions, and cp-split extensions, respectively. Similar remarks apply to mappingcylinders and mapping cones: for any morphism of extensions

I ��

˛

��

E ��

ˇ

��

Q

�

��

I 0 �� E 0 �� Q0,

(5)

we get extensions

Cyl.˛/ � Cyl.ˇ/ � Cyl.�/; Cone.˛/ � Cone.ˇ/ � Cone.�/I (6)

if the extensions in (5) are split or cp-split, so are the resulting extensions in (6).The familiar maps relating mapping cones and cylinders to cones and suspensions

continue to exist in our case. For any morphism f W A ! B in G-C�alg, we get amorphism of extensions

Sus.B/ ��

��

Cone.f / ��

��

A

Cone.B/ �� Cyl.f / �� A.

The bottom extension splits and the maps A $ Cyl.f / are inverse to each other upto homotopy. By naturality, the composite map Cone.f / ! A ! B factors throughCone.idB/ Š Cone.B/ and hence is homotopic to the zero map.

3 Universal functors with certain properties

When we study topological invariants for C�-algebras, we usually require homotopyinvariance and some exactness and stability conditions. Here we investigate theseconditions and their interplay and describe some universal functors.

We discuss homotopy invariant functors on G-C�alg and the homotopy categoryHo.G-C�alg/ in §3.1. This is parallel to classical topology. We turn to Morita–Rieffel invariance and C�-stability in §3.2–3.3. We describe the resulting localisationusing correspondences. By the way, in a C�-algebra context, stability usually refers toalgebras of compact operators instead of suspensions. §3.4 deals with various exactnessconditions: split-exactness, half-exactness, and additivity.

Whereas each of the above properties in itself seems rather weak, their combinationmay have striking consequences. For instance, a functor that is both C�-stable andsplit-exact is automatically homotopy invariant and satisfies Bott periodicity.

Throughout this section, we consider functors S! C where S is a full subcategoryof C�alg or G-C�alg for some locally compact group G. The target category Cmay be arbitrary in §3.1–§3.3; to discuss exactness properties, we require C to bean exact category or at least additive. Typical choices for S are the categories ofseparable or separable nuclear C�-algebras, or the subcategory of all separable nuclearG-C�-algebras with an amenable (or a proper) action of G.


Definition 26. Let P be a property for functors defined on S. A universal functorwith P is a functor u W S! UnivP .S/ such that

• xF ı u has P for each functor xF W UnivP .S/! C;

• any functor F W S! C with P factors uniquely as F D xF ı u for some functorxF W UnivP .S/! C.

Of course, a universal functor with P need not exist. If it does, then it restricts toa bijection between objects of S and UnivP .S/. Hence we can completely describeit by the sets of morphisms UnivP .A;B/ from A to B in UnivP .S/ and the mapsS.A;B/ ! UnivP .A;B/ for A;B 22 S; the universal property means that forany functor F W S ! C with P there is a unique functorial way to extend the mapsHomG.A;B/ ! C

�F.A/; F.B/

�to UnivP .A;B/. There is no a priori reason why

the morphism spaces UnivP .A;B/ for A;B 22 S should be independent of S; butthis happens in the cases we consider, under some assumption on S.

3.1 Homotopy invariance. The following discussion applies to any full subcategoryS � G-C�alg that is closed under the cylinder functor.

Definition 27. Let f0; f1 W A � B be two parallel morphisms in S. We write f0 � f1and call f0 and f1 homotopic if there is a homotopy between f0 and f1, that is, amorphism f W A! Cyl.B/ D C.Œ0; 1�; B/ with evt ı f D ft for t D 0; 1.

It is easy to check that homotopy is an equivalence relation on HomG.A;B/. We letŒA; B� be the set of equivalence classes. The composition of morphisms in S descendsto maps

ŒB; C � � ŒA; B�! ŒA; C �;�Œf �; Œg�

� 7! Œf ı g�;that is, f1 � f2 and g1 � g2 implies f1 ı f2 � g1 ı g2. Thus the sets ŒA; B� formthe morphism sets of a category, called homotopy category of S and denoted Ho.S/.The identity maps on objects and the canonical maps on morphisms define a canonicalfunctor S! Ho.S/. A morphism in S is called a homotopy equivalence if it becomesinvertible in Ho.S/.

Lemma 28. The following are equivalent for a functor F W S! C:

(a) F.ev0/ D F.ev1/ as maps F�C.Œ0; 1�; A/

�! F.A/ for all A 22 S;

(b) F.evt / induces isomorphisms F�C.Œ0; 1�; A/

�! F.A/ for allA 22 S, t 2 Œ0; 1�;(c) the embedding as constant functions const W A! C.Œ0; 1�; A/ induces an isomor-

phism F.A/! F�C.Œ0; 1�; A/

�for all A 22 S;

(d) F maps homotopy equivalences to isomorphisms;

(e) if two parallel morphisms f0; f1 W A � B are homotopic, then F.f0/ D F.f1/;(f) F factors through the canonical functor S! Ho.S/.

Furthermore, the factorisation Ho.S/! C in (f) is necessarily unique.

14 R. Meyer

Proof. We only mention two facts that are needed for the proof. First, we haveevt ı const D idA and const ı evt � idC.Œ0;1�;A/ for all A 22 S and all t 2 Œ0; 1�.Secondly, an isomorphism has a unique left and a unique right inverse, and these areagain isomorphisms.

The equivalence of (d) and (f) in Lemma 28 says that Ho.S/ is the localisation of Sat the family of homotopy equivalences.

Since C�Œ0; 1�;C0.X/

� Š C0.Œ0; 1� � X/ for any locally compact space X , ournotion of homotopy restricts to the usual one for pointed compact spaces. Hence theopposite of the homotopy category of pointed compact spaces is equivalent to a fullsubcategory of Ho.C�alg/.

The sets ŒA; B� inherit a base point Œ0� and a quotient topology from Hom.A;B/;thus Ho.S/ is enriched over pointed topological spaces as well. This topology onŒA; B� is not so useful, however, because it need not be Hausdorff.

A similar topology exists on Kasparov groups and can be defined in various ways,which turn out to be equivalent [12].

Let F W G-C�alg! H -C�alg be a functor with natural isomorphisms

F�C.Œ0; 1�; A/

� Š C.Œ0; 1�; F .A/�

that are compatible with evaluation maps for allA. The universal property implies thatFdescends to a functor Ho.G-C�alg/ ! Ho.H -C�alg/. In particular, this applies tothe suspension, cone, and cylinder functors and, more generally, to the functorsA˝max

andA˝min onG-C�alg for anyA 22 G-C�alg because both tensor product functorsare associative and commutative and

C.Œ0; 1�; A/ Š C.Œ0; 1�/˝max A Š C.Œ0; 1�/˝min A:

The same reasoning applies to the reduced and full crossed product functorsG-C�alg!C�alg.

We may stabilise the homotopy category with respect to the suspension functor andconsider a suspension-stable homotopy category with morphism spaces

fA;Bg WD lim�!k!1

ŒSuskA;SuskB�

for all A;B 22 G-C�alg. We may also enlarge the set of objects by adding formaldesuspensions and generalising the notion of spectrum. This is less interesting forC�-algebras than for spaces because most functors of interest satisfy Bott Periodicity,so that suspension and desuspension become equivalent.

3.2 Morita–Rieffel equivalence and stable isomorphism. One of the basic ideas ofnon-commutative geometry is that G Ë C0.X/ (or G Ër C0.X/) should be a substitutefor the quotient space GnX , which may have bad singularities. In the special case of afree and properG-spaceX , we expect thatG Ë C0.X/ and C0.GnX/ are “equivalent”


in a suitable sense. Already the simplest possible case X D G shows that we cannotexpect an isomorphism here because

G Ë C0.G/ Š G Ër C0.G/ Š K.L2G/:

The right notion of equivalence is a C�-version of Morita equivalence due to Marc A.Rieffel ([46], [47], [48]); therefore, we call it Morita–Rieffel equivalence.

The definition of Morita–Rieffel equivalence involves Hilbert modules over C�-al-gebras and the C�-algebras of compact operators on them; these notions are crucial forKasparov theory as well. We refer to [34] for the definition and a discussion of theirbasic properties.

Definition 29. Two G-C�-algebras A and B are called Morita–Rieffel equivalent ifthere are a fullG-equivariant Hilbert B-module E and aG-equivariant �-isomorphismK.E/ Š A.

It is possible (and desirable) to express this definition more symmetrically: E is anA;B-bimodule with two inner products taking values in A and B , satisfying variousconditions [46]. Morita–Rieffel equivalent G-C�-algebras have equivalent categoriesof G-equivariant Hilbert modules via E ˝B . The converse is unclear.

Example 30. The following is a more intricate example of a Morita–Rieffel equiva-lence. Let � and P be two subgroups of a locally compact group G. Then � acts onG=P by left translation and P acts on �nG by right translation. The correspondingorbit space is the double coset space�nG=P . Both�ËC0.G=P / andP ËC0.�nG/ arenon-commutative models for this double coset space. They are indeed Morita–Rieffelequivalent; the bimodule that implements the equivalence is a suitable completion ofCc.G/, the space of continuous functions with compact support on G.

These examples suggest that Morita–Rieffel equivalent C�-algebras describe thesame non-commutative space. Therefore, we expect that reasonable functors on C�algshould not distinguish between Morita–Rieffel equivalent C�-algebras. (We willslightly weaken this statement below.)

Definition 31. Two G-C�-algebras A and B are called stably isomorphic if there is aG-equivariant �-isomorphismA˝K.HG/ Š B˝K.HG/, where HG WD L2.G�N/is the direct sum of countably many copies of the regular representation ofG; we letGact on K.HG/ by conjugation, of course.

The following technical condition is often needed in connection with Morita–Rieffelequivalence.

Definition 32. A C�-algebra is called � -unital if it has a countable approximate identityor, equivalently, contains a strictly positive element.

Example 33. All separable C�-algebras and all unital C�-algebras are � -unital; thealgebra K.H / is � -unital if and only if H is separable.

16 R. Meyer

Theorem 34 ([7]). Two � -unital G-C�-algebras are G-equivariantly Morita–Rieffelequivalent if and only if they are stably isomorphic.

In the non-equivariant case, this theorem is due to Brown–Green–Rieffel [7]. Asimpler proof that carries over to the equivariant case appeared in [41].

3.3 C�-stable functors. The definition of C�-stability is more intuitive in the non-equivariant case:

Definition 35. Fix a rank-one projection p 2 K.`2N/. The resulting embeddingA! A˝K.`2N/, a 7! a˝ p, is called a corner embedding of A.

A functor F W C�alg! C is called C�-stable if any corner embedding induces anisomorphism F.A/ Š F �

A˝K.`2N/�.

The correct equivariant generalisation is the following:

Definition 36 ([36]). A functor F W G-C�alg! C is called C�-stable if the canonicalembeddings H1 ! H1 ˚H2 H2 induce isomorphisms

F�A˝K.H1/

� Š�! F�A˝K.H1 ˚H2/

� Š � F �A˝K.H2/

�

for all non-zero G-Hilbert spaces H1 and H2.

Of course, it suffices to require F�A ˝ K.H1/

� Š�! F�A ˝ K.H1 ˚H2/

�. It is

not hard to check that Definitions 35 and 36 are equivalent for trivial G.

Remark 37. We have argued in §3.2 why C�-stability is an essential property for anydecent homology theory for C�-algebras. Nevertheless, it is tempting to assume lessbecause C�-stability together with split-exactness has very strong implications.

One reasonable way to weaken C�-stability is to replace K.`2N/ by Mn for n 2 Nin Definition 35 (see [57]). If two unital C�-algebras are Morita–Rieffel equivalent,then they are also Morita equivalent as rings, that is, the equivalence is implementedby a finitely generated projective module. This implies that a matrix-stable functor isinvariant under Morita–Rieffel equivalence for unital C�-algebras.

Matrix-stability also makes good sense inG-C�alg for a compact groupG: simplyrequire H1 and H2 in Definition 36 to be finite-dimensional. But we seem to runinto problems for non-compact groups because they may have few finite-dimensionalrepresentations and we lack a finite-dimensional version of the equivariant stabilisationtheorem.

Our next goal is to describe the universal C�-stable functor. We abbreviate AK WDK.L2G/˝ A.

Definition 38. A correspondence from A to B (or A Ü B) is a G-equivariantHilbert BK-module E together with a G-equivariant essential (or non-degenerate)�-homomorphism f W AK ! K.E/.


Given correspondences E from A to B and F from B to C , their composition isthe correspondence from A to C with underlying Hilbert module E xBK F and mapAK ! K.E/ ! K.E xBK F /, where the last map sends T 7! T ˝ 1; this yieldscompact operators becauseBK maps to K.F /. See [34] for the definition of the relevantcompleted tensor product of Hilbert modules.

Up to isomorphism, the composition of correspondences is associative and the iden-tity mapsA! A D K.A/ act as unit elements. Hence we get a category CorrG whosemorphisms are the isomorphism classes of correspondences. It may have advantagesto treat CorrG as a 2-category.

Any �-homomorphism ' W A! B yields a correspondence f W A! K.E/ from A

to B , so that we get a canonical functor \ W G-C�alg! CorrG . We let E be the rightideal '.AK/ � BK in BK, viewed as a Hilbert B-module. Then f .a/ � b WD '.a/ � brestricts to a compact operator f .a/ on E and f W A ! K.E/ is essential. It can bechecked that this construction is functorial.

In the following proposition, we require that the category of G-C�-algebras S beclosed under Morita–Rieffel equivalence and consist of � -unital G-C�-algebras. Welet CorrS be the full subcategory of CorrG with object class S.

Proposition 39. The functor \ W S! CorrS is the universal C�-stable functor on S;that is, it is C�-stable, and any other such functor factors uniquely through \.

Proof. First we sketch the proof in the non-equivariant case.We verify that \ is C�-stable. The Morita–Rieffel equivalence between K.`2N/˝

A Š K�`2.N; A/

�and A is implemented by the Hilbert module `2.N; A/, which

yields a correspondence�id; `2.N; A/

�from K.`2N/˝ A to A; this is inverse to the

correspondence induced by a corner embedding A! K.`2N/˝ A.A HilbertB-module E with an essential �-homomorphismA! K.E/ is countably

generated because A is assumed � -unital. Kasparov’s Stabilisation Theorem yields anisometric embedding E ! `2.N; B/. Hence we get �-homomorphisms

A! K.`2N/˝ B B:

This diagram induces a map F.A/ ! F.K.`2N/ ˝ B/ Š F.B/ for any C�-stablefunctor F . Now we should check that this well-defines a functor xF W CorrS ! C withxF ı \ D F , and that this yields the only such functor. We omit these computations.

The generalisation to the equivariant case uses the crucial property of the left regularrepresentation that L2.G/˝H Š L2.G �N/ for any countably infinite-dimensionalG-Hilbert space H . Since we replace A and B by AK and BK in the definition ofcorrespondence right away, we can use this to repair a possible lack ofG-equivariance;similar ideas appear in [36].

Example 40. Let u be a G-invariant multiplier of B with u�u D 1; such u arealso called isometries. Then b 7! ubu� defines a �-homomorphism B ! B . Theresulting correspondence B Ü B is isomorphic as a correspondence to the identity

18 R. Meyer

correspondence: the isomorphism is given by left multiplication with u, which definesa G-equivariant unitary operator from B to the closure of uBu� � B D u � B .

Hence inner endomorphisms act trivially on C�-stable functors. Actually, this isone of the computations that we have omitted in the proof above; the argument can befound in [11].

Now we make the definition of a correspondence more concrete if A is unital. Wehave an essential �-homomorphism ' W A ! K.E/ for some G-equivariant HilbertB-module E . Since A is unital, this means that K.E/ is unital and ' is a unital�-homomorphism. Then E is finitely generated. Thus E D B1 �p for some projectionp 2 M1.B/ and ' is a �-homomorphism ' W A ! M1.B/ with '.1/ D p. Two�-homomorphisms '1; '2 W A � M1.B/ yield isomorphic correspondences if andonly if there is a partial isometryv 2M1.B/withv'1.x/v� D '2.x/ andv�'2.a/v D'1.a/ for all a 2 A.

Finally, we combine homotopy invariance and C�-stability and consider the univer-sal C�-stable homotopy-invariant functor. This functor is much easier to characterise:the morphisms in the resulting universal category are simply the homotopy classes ofG-equivariant �-homomorphisms K

�L2.G �N/

�˝ A! K�L2.G �N/

�˝ B/ (see[36], Proposition 6.1). Alternatively, we get the same category if we use homotopyclasses of correspondences A Ü B instead.

3.4 Exactnessproperties. Throughout this subsection, we consider functorsF W S!C with values in an exact category C. If C is merely additive to begin with, we canequip it with the trivial exact category structure for which all extensions split. We alsosuppose that S is closed under the kinds of C�-algebra extensions that we consider;depending on the notion of exactness, this means: direct product extensions, split ex-tensions, cp-split extensions, or all extensions, respectively. Recall that split extensionsin G-C�alg are required to split by a G-equivariant �-homomorphism.

3.4.1 Additive functors. The most trivial split extensions inG-C�alg are the productextensions A � A � B � B for two objects A;B . In this case, the coordinateembeddings and projections provide maps

A � A � B � B: (7)

Definition 41. We call F additive if it maps product diagrams (7) in S to direct sumdiagrams in C.

There is a partially defined addition on �-homomorphisms: call two parallel �-homomorphisms'; W A � B orthogonal if '.a1/ � .a2/ D 0 for all a1; a2 2 A. Equivalently,' C W a 7! '.a/C .a/ is again a �-homomorphism.

Lemma 42. The functor F is additive if and only if, for all A;B 22 S, the mapsHom.A;B/ ! C

�F.A/; F.B/

�satisfy F.' C / D F.'/ C F. / for all pairs of

orthogonal parallel �-homomorphisms '; .


Alternatively, we may also require additivity for coproducts (that is, free products).Of course, this only makes sense if S is closed under coproducts in G-C�alg. Thecoproduct A t B in G-C�alg comes with canonical maps A � A t B � B as well;the maps �A W A ! A t B and �B W B ! A t B are the coordinate embeddings, themaps �A W A t B ! A and �B W A t B ! B restrict to .idA; 0/ and .0; idB/ on Aand B , respectively.

Definition 43. We call F additive on coproducts if it maps coproduct diagrams A �A t B � B to direct sum diagrams in C.

The coproduct and product are related by a canonicalG-equivariant �-homomorphism' W A t B � A � B that is compatible with the maps to and from A and B , that is,' ı �A D �A, �A ı ' D �A, and similarly for B . There is no map backwards, but thereis a correspondence W A � B Ü A t B , which is induced by the G-equivariant�-homomorphism

A � B !M2.A t B/; .a; b/ 7!��A.a/ 0

0 �B.b/

�:

It is easy to see that the composite correspondence ' ı is equal to the identitycorrespondence onA�B . The other composite ı' is not the identity correspondence,but it is homotopic to it (see [9], [10]). This yields:

Proposition 44. If F is C�-stable and homotopy invariant, then the canonical mapF.'/ W F.A t B/ ! F.A � B/ is invertible. Therefore, additivity and additivity forcoproducts are equivalent for such functors.

The correspondence exists because the stabilisation creates enough room to re-place �A and �B by homotopic homomorphisms with orthogonal ranges. We can achievethe same effect by a suspension (shift �A and �B to the open intervals .0; 1=2/ and .1=2; 1/,respectively). Therefore, any homotopy invariant functor satisfies F

�Sus.A t B/� Š

F�Sus.A � B/�.

3.4.2 Split-exact functors

Definition 45. We call F split-exact if, for any split extension Ki� E

p� Q with

section s W Q! E, the map�F.i/; F.s/

� W F.K/˚ F.Q/! F.E/ is invertible.

It is clear that split-exact functors are additive.Split-exactness is useful because of the following construction of Joachim Cuntz [9].Let B G E be a G-invariant ideal and let fC; f� W A � E be G-equivariant

�-homomorphisms with fC.a/ � f�.a/ 2 B for all a 2 A. Equivalently, fC and f�both lift the same morphism Nf W A ! E=B . The data .A; fC; f�; E; B/ is called aquasi-homomorphism from A to B .

Pulling back the extension B � E � E=B along Nf , we get an extension B �E 0 � A with two sections f 0C; f 0� W A � E 0. The split-exactness of F shows that

20 R. Meyer

F.B/ � F.E 0/ � F.A/ is a split extension in C. Since both F.f 0�/ and F.f 0C/are sections for it, we get a map F.f 0C/ � F.f 0�/ W F.A/ ! F.B/. Thus a quasi-homomorphism induces a mapF.A/! F.B/ ifF is split-exact. The formal propertiesof this construction are summarised in [11].

Given a C�-algebra A, there is a universal quasi-homomorphism out of A. LetQ.A/ WD A t A be the coproduct of two copies of A and let �A W Q.A/ ! A be thefolding homomorphism that restricts to idA on both factors. Let q.A/ be its kernel.The two canonical embeddings A ! A t A are sections for the folding homomor-phism. Hence we get a quasi-homomorphism A � Q.A/ F q.A/. The universalproperty of the free product shows that any quasi-homomorphism yields a G-equivari-ant �-homomorphism q.A/! B .

Theorem 46. Suppose S is closed under split extensions and tensor products withC.Œ0; 1�/ and K.`2N/. If F W S! C is C�-stable and split-exact, then F is homotopyinvariant.

This is a deep result of Nigel Higson [24]; a simple proof can be found in [11].Besides basic properties of quasi-homomorphisms, it uses that inner endomorphismsact identically on C�-stable functors (Example 40).

Actually, the literature only contains Theorem 46 for functors on C�alg. But theproof in [11] works for functors on categories S as above.

3.4.3 Exact functors

Definition 47. We call F exact if F.K/ ! F.E/ ! F.Q/ is exact (at F.E/) forany extension K � E � Q in S. More generally, given a class E of extensionsin S like, say, the class of equivariantly cp-split extensions, we define exactness forextensions in E .

It is easy to see that exact functors are additive.Most functors we are interested in satisfy homotopy invariance and Bott periodicity,

and these two properties prevent a non-zero functor from being exact in the strongersense of being left or right exact. This explains why our notion of exactness is muchweaker than usual in homological algebra.

It is reasonable to require that a functor be part of a homology theory, that is,a sequence of functors .Fn/n2Z together with natural long exact sequences for allextensions [54]. We do not require this additional information because it tends to behard to get a priori but often comes for free a posteriori:

Proposition 48. Suppose that F is homotopy invariant and exact (or exact for equiv-ariantly cp-split extensions). Then F has long exact sequences of the form

� � � ! F�Sus.K/

�! F�Sus.E/

�! F�Sus.Q/

�! F.K/! F.E/! F.Q/

for any (equivariantly cp-split) extension K � E � Q. In particular, F is split-exact.


See §21.4 in [5] for the proof.There probably exist exact functors that are not split-exact. It is likely that the

algebraic K1-functor provides a counterexample: it is exact but not split-exact onthe category of rings [49]; but I do not know a counterexample to its split-exactnessinvolving only C�-algebras.

Proposition 48 and Bott periodicity yield long exact sequences that are infinite inboth directions. Thus an exact homotopy invariant functor that satisfies Bott periodicityis part of a homology theory in a canonical way.

For universal constructions, we should replace a single functor by a homologytheory, that is, a sequence of functors. The universal functors in this context are non-stable versions of E-theory and KK-theory. We refer to [27] for details.

A weaker property than exactness is the existence of Puppe exact sequences formapping cones. The Puppe exact sequence is the special case of the long exact sequenceof Proposition 48 for extensions of the form Sus.B/ � Cone.f / � A for a morphismf W A ! B . In practice, the exactness of a functor is often established by reducing it

to the Puppe exact sequence. Let Ki� E

p� Q be an extension. The Puppe exact

sequence yields the long exact sequence for the extension Cone.p/ � Cyl.p/ � Q.There is a canonical morphism of extensions

K ��

��

E ��

��

Q

Cone.p/ �� Cyl.p/ �� Q,

where the vertical map E ! Cyl.p/ is a homotopy equivalence. Hence a functor withPuppe exact sequences is exact for K � E � Q if and only if it maps the verticalmap K ! Cone.p/ to an isomorphism.

4 Kasparov theory

We define KKG as the universal split-exact C�-stable functor onG-C�sep; since split-exact and C�-stable functors are automatically homotopy invariant, KKG is the univer-sal split-exact C�-stable homotopy functor as well. The universal property of Kasparovtheory due to Cuntz and Higson asserts that this is equivalent to Kasparov’s definition.We examine some basic properties of Kasparov theory and, in particular, show how toget functors between Kasparov categories.

We let EG be the universal exact C�-stable homotopy functor on G-C�sep or,equivalently, the universal exact, split-exact, and C�-stable functor.

Kasparov’s own definition of his theory is inspired by previous work of Atiyah [1]on K-homology; later, he also interacted with the work of Brown–Douglas–Fillmore[6] on extensions of C�-algebras. A construction in abstract homotopy theory providesa homology theory for spaces that is dual to K-theory. Atiyah realized that certainabstract elliptic differential operators provide cycles for this dual theory; but he did

22 R. Meyer

not know the equivalence relation to put on these cycles. Brown–Douglas–Fillmorestudied extensions of C0.X/ (and more general C�-algebras) by the compact operatorsand found that the resulting structure set is naturally isomorphic to a K-homology group.

Kasparov unified and vastly generalised these two results, defining a bivariant func-tor KK�.A;B/ that combines K-theory and K-homology and that is closely related tothe classification of extensions B˝K � E � A (see [29], [30]). A deep theorem ofKasparov shows that two reasonable equivalence relations for these cycles coincide; thisclarifies the homotopy invariance of the extension groups of Brown–Douglas–Fillmore.Furthermore, he constructed an equivariant version of his theory in [31] and applied itto prove the Novikov conjecture for discrete subgroups of Lie groups.

The most remarkable feature of Kasparov theory is an associative product on KKcalled Kasparov product. This generalises various known product constructions inK-theory and K-homology and allows to view KK as a category.

In applications, we usually need some non-obvious KK-element, and we mustcompute certain Kasparov products explicitly. This requires a concrete description ofKasparov cycles and their products. Since both are somewhat technical, we do notdiscuss them here and merely refer to [5] for a detailed treatment and to [56] for a veryuseful survey article. Instead, we use Higson’s characterisation of KK by a universalproperty [23], which is based on ideas of Cuntz ([10], [9]). The extension to theequivariant case is due to Thomsen [58]. A simpler proof of Thomsen’s theorem andvarious related results can be found in [36].

We do not discuss KKG for Z=2-gradedG-C�-algebras here because it does not fit sowell with the universal property approach, which would simply yield KKG�Z=2 becauseZ=2-graded G-C�-algebras are the same as G � Z=2-C�-algebras. The relationshipbetween the two theories is explained in [36], following Ulrich Haag [22]. The gradedversion of Kasparov theory is often useful because it allows us to treat even and oddKK-cycles simultaneously.

Fix a locally compact group G. The Kasparov groups KKG0 .A;B/ for A;B 2G-C�sep form the morphism sets A ! B of a category, which we denote by KKG ;the composition in KKG is the Kasparov product. The categories G-C�sep and KKG

have the same objects. We have a canonical functor

KKG W G-C�sep! KKG

that acts identically on objects. This functor contains all information about equivariantKasparov theory for G.

Definition 49. A G-equivariant �-homomorphism f W A ! B is called a KKG-equivalence if KKG.f / is invertible in KKG .

Theorem 50. Let S be a full subcategory ofG-C�sep that is closed under suspensions,G-equivariantly cp-split extensions, and Morita–Rieffel equivalence. Let KKG.S/ bethe full subcategory of KKG with object class S and let KKGS W S! KKG.S/ be therestriction of KKG .


The functor KKGS W S ! KKGS is the universal split-exact C�-stable functor; in

particular, KKG.S/ is an additive category. In addition, it has the following propertiesand is, therefore, universal among functors on S with some of these extra properties:

• it is homotopy invariant;

• it is exact for G-equivariantly cp-split extensions;

• it satisfies Bott periodicity, that is, in KKG there are natural isomorphismsSus2.A/ Š A for all A 22 KKG .

Corollary 51. Let F W S! C be split-exact and C�-stable. Then F factors uniquelythroughKKGS, is homotopy invariant, and satisfiesBott periodicity. AKKG-equivalenceA! B in S induces an isomorphism F.A/! F.B/.

We will view the universal property of Theorem 50 as a definition of KKG and thusof the groups KKG0 .A;B/. We also let

KKGn .A;B/ WD KKG�A;Susn.B/

�Isince the Bott periodicity isomorphism identifies KKG2 Š KKG0 , this yields a Z=2-gra-ded theory.

Now we describe KKG0 .A;B/ more concretely. Recall AK WD A˝K.L2G/.

Proposition 52. Let A and B be two G-C�-algebras. There is a natural bijectionbetween the morphism sets KKG0 .A;B/ in KKG and the set Œq.AK/; BK˝K.`2N/� ofhomotopy classes of G-equivariant �-homomorphisms from q.AK/ to BK ˝K.`2N/.

Proof. The canonical functor G-C�sep ! KKG is C�-stable and split-exact, andtherefore homotopy invariant by Theorem 46 (this is already asserted in Theorem 50).Proposition 44 yields that it is additive for coproducts. Split-exactness for the splitextension q.A/ � Q.A/ � A shows that idA � 0 W Q.A/ ! A restricts to a KKG-equivalence q.A/ � A. Similarly, C�-stability yields KKG-equivalences A � AK andB � BK ˝ K.`2N/. Hence homotopy classes of �-homomorphisms from q.AK/ toBK˝K.`2N/ yield classes in KKG0 .A;B/. Using the concrete description of Kasparovcycles, which we have not discussed, it is checked in [36] that this map yields a bijectionas asserted.

Another equivalent description is

KKG0 .A;B/ Š Œq.AK/˝K.`2N/; q.BK/˝K.`2N/�Iin this approach, the Kasparov product becomes simply the composition of morphisms.Proposition 52 suggests that q.AK/ andBK˝K.`2N/may be the cofibrant and fibrantreplacement of A and B in some model category related to KKG . But it is not clearwhether this is the case. The model category structure constructed in [28] is certainlyquite different.

24 R. Meyer

By the universal property, K-theory descends to a functor on KK, that is, we getcanonical maps

KK0.A;B/! Hom�K�.A/;K�.B/

�for all separable C�-algebras A;B , where the right hand side denotes grading-pre-serving group homomorphisms. For A D C, this yields a map KK0.C; B/ !Hom

�Z;K0.B/

� Š K0.B/. Using suspensions, we also get a corresponding mapKK1.C; B/! K1.B/.

Theorem 53. The maps KK�.C; B/ ! K�.B/ constructed above are isomorphismsfor all B 22 C�sep.

Thus Kasparov theory is a bivariant generalisation of K-theory. Roughly speaking,KK�.A;B/ is the place where maps between K-theory groups live. Most construc-tions of such maps, say, in index theory can, in fact, be improved to yield elements ofKK�.A;B/. One reason why this has to be so is the Universal Coefficient Theorem(UCT), which computes KK�.A;B/ from K�.A/ and K�.B/ for many C�-algebrasA;B . If A satisfies the UCT, then any grading preserving group homomorphismK�.A/! K�.B/ lifts to an element of KK0.A;B/.

4.1 Extending functors and identities to KKG . We can use the universal propertyto extend various functors G-C�sep ! H -C�sep to functors KKG ! KKH . Weexplain this by an example:

Proposition 54. The full and reduced crossed product functors

G Ër ; G Ë W G-C�alg! C�alg

extend to functors G Ër ; G Ë W KKG ! KK called descent functors.

Gennadi Kasparov [31] constructs these functors directly using the concrete descrip-tion of Kasparov cycles. This requires a certain amount of work; in particular, checkingfunctoriality involves knowing how to compute Kasparov products. The constructionvia the universal property is formal:

Proof. We only write down the argument for reduced crossed products, the other caseis similar. It is well-known that G Ër

�A ˝ K.H /

� Š .G Ër A/ ˝ K.H / for anyG-Hilbert space H . Therefore, the composite functor

G-C�sepGËr��! C�sep

KK��! KK

is C�-stable. Proposition 9 shows that this functor is split-exact as well (regardless ofwhether G is an exact group). Now the universal property provides an extension to afunctor KKG ! KK.

Similarly, we get functors

A˝min ; A˝max W KKG ! KKG


for any G-C�-algebra A. Since these extensions are natural, we even get bifunctors

˝min;˝max W KKG �KKG ! KKG :

The associativity, commutativity, and unit constraints in G-C�alg induce correspond-ing constraints in KKG , so that both ˝min and ˝max turn KKG into a symmetricmonoidal category.

Another example is the functor � W C�alg ! G-C�alg that equips a C�-algebrawith the trivial G-action; it extends to a functor � W KK! KKG .

The universal property also allows us to prove identities between functors. Forinstance, we have natural isomorphisms G Ër .�.A/˝min B/ D A˝min .G Ër B/ forall G-C�-algebras B . Naturality means, to begin with, that the diagram

G Ër .�.A1/˝min B1/Š ��

GËr�.˛/˝minˇ

��

A1 ˝min .G Ër B1/

˛˝min.GËrˇ/

��

G Ër .�.A2/˝min B2/Š �� A2 ˝min .G Ër B2/

commutes if ˛ W A1 ! A2 and ˇ W B1 ! B2 are a �-homomorphism and a G-equi-variant �-homomorphism, respectively. Two applications of the uniqueness part ofthe universal property show that this diagram remains commutative in KK if ˛ 2KK0.A1; A2/ and ˇ 2 KKG0 .B1; B2/. Similar remarks apply to the natural iso-morphism G Ë .�.A/ ˝max B/ Š A ˝max .G Ë B/ and hence to the isomorphismsG Ë �.A/ Š C�.G/˝max A and G Ër �.A/ Š C�red.G/˝min A.

Adjointness relations in Kasparov theory are usually proved most easily by con-structing the unit and counit of the adjunction. For instance, if G is a compact groupthen the functor � is left adjoint to G Ë D G Ër , that is, for all A 22 KK andB 22 KKG , we have natural isomorphisms

KKG� .�.A/; B/ Š KK�.A;G Ë B/: (8)

This is also known as the Green–Julg Theorem. For A D C, it specialises to a naturalisomorphism KG� .B/ Š K�.G Ë B/; this was one of the first appearances of non-commutative algebras in topological K-theory.

Proof of (8). We already know that � and G Ë are functors between KK and KKG .It remains to construct natural elements

˛A 2 KK0�A;G Ë �.A/

�; ˇB 2 KKG0 .�.G Ë B/;B/

for allA 22 KK,B 22 KKG that satisfy the conditions for unit and counit of adjunction[35].

The main point is that �.G Ë B/ is the G-fixed point subalgebra of BK D B ˝K.L2G/. The embedding �.G Ë B/ ! BK provides a G-equivariant correspon-dence ˇB from �.G Ë B/ to B and thus an element of KKG0 .�.G Ë B/;B/. This

26 R. Meyer

construction is certainly natural for G-equivariant �-homomorphisms and hence forKKG-morphisms by the uniqueness part of the universal property of KKG .

Let e� W C ! C�.G/ be the embedding that corresponds to the trivial representationof G. Recall that G Ë �.A/ Š C�.G/˝ A. Hence the exterior product of the identitymap on A and KK.e� / provides ˛A 2 KK0

�A;G Ë �.A/

�. Again, naturality for

�-homomorphisms is clear and implies naturality for morphisms in KK.Finally, it remains to check that

�.A/�.˛A/��! �

�G Ë �.A/

� ˇ�.A/��! �.A/;

G Ë B ˛GËB��! G Ë �.G Ë B/GËˇB��! G Ë B

are the identity morphisms in KKG . Then we get the desired adjointness using a generalconstruction from category theory (see [35]). In fact, both composites are equal to theidentity already as correspondences, so that we do not have to know anything aboutKasparov theory except its C�-stability to check this.

A similar argument yields an adjointness relation

KKG0�A; �.B/

� Š KK0.G Ë A;B/ (9)

for a discrete groupG. More conceptually, (9) corresponds via Baaj–Skandalis duality[3] to the Green–Julg Theorem for the dual quantum group of G, which is compactbecauseG is discrete. But we can also write down unit and counit of adjunction directly.

The trivial representation C�.G/! C yields natural �-homomorphisms

G Ë �.B/ Š C�.G/˝max B ! B

and hence ˇB 2 KK0.G Ë �.B/; B/. The canonical embedding A ! G Ë A isG-equivariant if we let G act on G Ë A by conjugation; but this action is inner, sothat G Ë A and �.G Ë A/ are G-equivariantly Morita–Rieffel equivalent. Thus thecanonical embedding A ! G Ë A yields a correspondence A Ü �.G Ë A/ and˛A 2 KKG0

�A; �.G Ë A/

�. We must check that the composites

G Ë A GË˛A��! G Ë �.G Ë A/ˇGËA��! G Ë A;

�.B/˛�.B/��! �

�G Ë �.B/

� �.ˇB /��! �.B/

are identity morphisms in KK and KKG , respectively. Once again, this holds alreadyon the level of correspondences.

4.2 Triangulated category structure. We turn KKG into a triangulated category byextending standard constructions for topological spaces [38]. Some arrows changedirection because the functor C0 from spaces to C�-algebras is contravariant. We havealready observed that KKG is additive. The suspension is given by†�1.A/ WD Sus.A/.Since Sus2.A/ Š A in KKG by Bott periodicity, we have † D †�1. Thus we do notneed formal desuspensions as for the stable homotopy category.


Definition 55. A triangle A ! B ! C ! †A in KKG is called exact if it isisomorphic as a triangle to the mapping cone triangle

Sus.B/! Cone.f /! Af�! B

for some G-equivariant �-homomorphism f .

Alternatively, we can use G-equivariantly cp-split extensions in G-C�sep. Anysuch extension I � E � Q determines a class in KKG1 .Q; I / Š KKG0 .Sus.Q/; I /,so that we get a triangle Sus.Q/! I ! E ! Q in KKG . Such triangles are calledextension triangles. A triangle in KKG is exact if and only if it is isomorphic to theextension triangle of a G-equivariantly cp-split extension [38].

Theorem 56. With the suspension automorphism and exact triangles defined above,KKG is a triangulated category. So is KKG.S/ if S � G-C�sep is closed undersuspensions,G-equivariantly cp-split extensions, and Morita–Rieffel equivalence as inTheorem 50.

Proof. That KKG is triangulated is proved in detail in [38]. We do not discuss thetriangulated category axioms here. Most of them amount to properties of mappingcone triangles that can be checked by copying the corresponding arguments for thestable homotopy category (and reverting arrows). These axioms hold for KKG.S/because they hold for KKG . The only axiom that requires more care is the existenceaxiom for exact triangles; it requires any morphism to be part of an exact triangle. Wecan prove this as in [38] using the concrete description of KKG0 .A;B/ in Proposition 52.For some applications like the generalisation to KKG.S/, it is better to use extensiontriangles instead. Any f 2 KKG0 .A;B/ Š KKG1 .Sus.A/; B/ can be represented bya G-equivariantly cp-split extension K.HB/ � E � Sus.A/, where HB is a fullG-equivariant Hilbert B-module, so that K.HB/ is G-equivariantly Morita–Rieffelequivalent to B . The extension triangle of this extension contains f and belongs toKKG.S/ by our assumptions on S.

Since model category structures related to C�-algebras are rather hard to get (com-pare [28]), triangulated categories seem to provide the most promising formal setupfor extending results from classical spaces to C�-algebras. An earlier attempt can befound in [54]. Triangulated categories clarify the basic bookkeeping with long exactsequences. Mayer–Vietoris exact sequences and inductive limits are discussed fromthis point of view in [38]. More importantly, this framework sheds light on more ad-vanced constructions like the Baum–Connes assembly map. We will briefly discussthis below.

4.3 The Universal Coefficient Theorem. There is a very close relationship betweenK-theory and Kasparov theory. We have already seen that K�.A/ Š KK�.C; A/ isa special case of KK. Furthermore, KK inherits deep properties of K-theory such asBott periodicity. Thus we may hope to express KK�.A;B/ using only the K-theory of

28 R. Meyer

A and B – at least for many A and B . This is the point of the Universal CoefficientTheorem.

The Kasparov product provides a canonical homomorphism of graded groups

� W KK�.A;B/! Hom��K�.A/;K�.B/

�;

where Hom� denotes the Z=2-graded Abelian group of all group homomorphismsK�.A/ ! K�.B/. There are topological reasons why � cannot always be invertible:since Hom� is not exact, the bifunctor Hom�

�K�.A/;K�.B/

�would not be exact on

cp-split extensions. A construction of Lawrence Brown provides another natural map

W ker � ! Ext��K�C1.A/;K�.B/

�:

The following theorem is due to Jonathan Rosenberg and Claude Schochet [53], [51];see also [5].

Theorem 57. The following are equivalent for a separable C�-algebra A:

(a) KK�.A;B/ D 0 for all B 22 KK with K�.B/ D 0;(b) the map � is surjective and is bijective for all B 22 KK;

(c) for all B 22 KK, there is a short exact sequence of Z=2-graded Abelian groups

Ext��K�C1.A/;K�.B/

�� KK�.A;B/ � Hom�

�K�.A/;K�.B/

�:

(d) A belongs to the smallest class of C�-algebras that contains C and is closed underKK-equivalence, suspensions, countable direct sums, and cp-split extensions;

(e) A is KK-equivalent to C0.X/ for some pointed compact metrisable space X .

If these conditions are satisfied, then the extension in (c) is natural and splits, but thesection is not natural.

The class of C�-algebras with these properties is also called the bootstrap classbecause of description (d). Alternatively, we may say that they satisfy the UniversalCoefficient Theorem because of (c). Since commutative C�-algebras are nuclear, (e)implies that the natural map A ˝max B ! A ˝min B is a KK-equivalence if A or Bbelongs to the bootstrap class [55]. This fails for some A, so that the Universal Coeffi-cient Theorem does not hold for all A. Remarkably, this is the only obstruction to theUniversal Coefficient Theorem known at the moment: we know no nuclear C�-algebrathat does not satisfy the Universal Coefficient Theorem. As a result, we can expressKK�.A;B/ using only K�.A/ and K�.B/ for many A and B .

When we restrict attention to nuclear C�-algebras, then the bootstrap class is closedunder various operations like tensor products, arbitrary extensions and inductive lim-its (without requiring any cp-sections), and under crossed products by torsion-freeamenable groups. Remarkably, there are no general results about crossed products byfinite groups.


The Universal Coefficient Theorem and the universal property of KK imply thatvery few homology theories for (pointed compact metrisable) spaces can extend to thenon-commutative setting. More precisely, if we require the extension to be split-exact,C�-stable, and additive for countable direct sums, then only K-theory with coefficientsis possible. Thus we rule out most of the difficult (and interesting) problems in stablehomotopy theory. But if we only want to study K-theory, anyway, then the operatoralgebraic framework usually provides very good analytical tools. This is most valuablefor equivariant generalisations of K-theory.

Jonathan Rosenberg and Claude Schochet [50] have also constructed a spectralsequence that, in favourable cases, computes KKG.A;B/ from KG� .A/ and KG� .B/;they require G to be a compact Lie group with torsion-free fundamental group andA and B to belong to a suitable bootstrap class. This equivariant UCT is clarifiedin [39], [40].

4.4 E-theory and asymptotic morphisms. Recall that Kasparov theory is only exactfor (equivariantly) cp-split extensions. E-theory is a similar theory that is exact for allextensions.

Definition 58. We let EG W G-C�sep! EG be the universal C�-stable, exact homo-topy functor.

Lemma 59. The functor EG is split-exact and factors through KKG W G-C�sep !KKG . Hence it satisfies Bott periodicity.

Proof. Proposition 48 shows that any exact homotopy functor is split-exact. The re-maining assertions now follow from Corollary 51.

The functor E (for trivialG) was first defined as above by Nigel Higson [25]. ThenAlain Connes and Nigel Higson [8] found a more concrete description using asymptoticmorphisms. This is what made the theory usable. The equivariant generalisation of thetheory is due to Erik Guentner, Nigel Higson, and Jody Trout [21].

We write EGn .A;B/ for the space of morphisms A ! Susn.B/ in EG . Bott peri-odicity shows that there are only two different groups to consider.

Definition 60. The asymptotic algebra of a C�-algebra B is the C�-algebra

Asymp.B/ WD Cb.RC; B/=C0.RC; B/:

An asymptotic morphism A! B is a �-homomorphism f W A! Asymp.B/.

Representing elements of Asymp.B/ by bounded functions Œ0;1/ ! B , we canrepresent f by a family of maps ft W A! B such that ft .a/ 2 Cb.RC; B/ for each a 2A and the mapa 7! ft .a/ satisfies the conditions for a �-homomorphism asymptoticallyfor t !1. This provides a concrete description of asymptotic morphisms and explainsthe name.

If a locally compact group G acts on B , then Asymp.B/ inherits an action of G bynaturality (which need not be strongly continuous).

30 R. Meyer

Definition 61. Let A and B be two G-C�-algebras for a locally compact group G. AG-equivariant asymptotic morphism fromA toB is aG-equivariant �-homomorphismf W A ! Asymp.B/. We write �A;B� for the set of homotopy classes of G-equi-variant asymptotic morphisms from A to B . Here a homotopy is a G-equivariant�-homomorphism A! Asymp

�C.Œ0; 1�; B/

�.

The asymptotic algebra fits, by definition, into an extension

C0.RC; B/ � Cb.RC; B/ � Asymp.B/:

Notice that C0.RC; B/ Š Cone.B/ is contractible. If f W A ! Asymp.B/ is aG-equivariant asymptotic morphism, then we can use it to pull back this extensionto an extension Cone.B/ � E � A inG-C�alg; theG-action onE is automaticallystrongly continuous. If F is exact and homotopy invariant, then F

�Susn.E/

� !F

�Susn.A/

�is an isomorphism for all n � 1 by Proposition 48. The evaluation

map Cb.RC; B/ ! B at some t 2 RC pulls back to a morphism E ! B , andthese morphisms for different t are all homotopic. Hence we get a well-defined mapF

�Susn.A/

� Š F �Susn.E/

�! F�Susn.B/

�for each asymptotic morphism A! B .

This explains how asymptotic morphisms are related to exact homotopy functors. Thisobservation leads to the following theorem:

Theorem 62. There are natural bijections

EG0 .A;B/ Š�

Sus�AK ˝K.`2N/

�;Sus

�BK ˝K.`2N/

��for all separable G-C�-algebras A;B .

An important step in the proof of Theorem 62 is the Connes–Higson construction,which to an extension I � E � Q in C�sep associates an asymptotic morphismSus.Q/! I . A G-equivariant generalisation of this construction is discussed in [59].Thus any extension inG-C�sep gives rise to an exact triangle Sus.Q/! I ! E ! Q

in EG .This also leads to the triangulated category structure of EG . As for KKG , we can

define it using mapping cone triangles or extension triangles – both approaches yieldthe same class of exact triangles. The canonical functor KKG ! EG is exact becauseit evidently preserves mapping cone triangles.

Now that we have two bivariant homology theories with apparently very similarformal properties, we must ask which one we should use. It may seem that the betterexactness properties of E-theory raise it above KK-theory. But actually, these strongexactness properties have a drawback: for a general group G, the reduced crossedproduct functor need not be exact, so that there is no guarantee that it extends to afunctor EG ! E. Only full crossed products exist for all groups by Proposition 6; theconstruction ofG Ë W EG ! E is the same as in KK-theory. Similar problems occurwith˝min but not with˝max.

Furthermore, since KKG has a weaker universal property, it acts on more functors,so that results about KKG have stronger consequences. A good example of a functor


that is split-exact but probably not exact is local cyclic cohomology (see [37], [45]).Therefore, the best practice seems to prove results in KKG if possible.

Many applications can be done with either E or KK, we hardly notice any difference.An explanation for this is the work of Houghton-Larsen and Thomsen ([27], [59]),which describes KKG0 .A;B/ in the framework of asymptotic morphisms. Recall thatasymptotic morphisms A ! B generate extensions Cone.B/ � E � A. If thisextension is G-equivariantly cp-split, then the projection map E � A is a KKG-equivalence. AG-equivariant completely positive contractive section for the extensionexists if and only if we can represent our asymptotic morphism by a continuous familyof G-equivariant, completely positive contractions ft W A! B , t 2 Œ0;1/.Definition 63. Let �A;B�cp be the set of homotopy classes of asymptotic morphismsfrom A to B that can be lifted to aG-equivariant, completely positive, contractive mapA! Cb.RC; B/; of course, we only use homotopies with the same kind of lifting.

Theorem 64 ([59]). There are natural bijections

KKG0 .A;B/ Š�

Sus�AK ˝K.`2N/

�;Sus

�BK ˝K.`2N/

��cp

for all separableG-C�-algebrasA;B; the canonical functor KKG ! EG correspondsto the obvious map that forgets the additional constraints.

Corollary 65. If A is a nuclear C�-algebra, then KK�.A;B/ Š E�.A;B/.

Proof. The Choi–Effros Lifting Theorem asserts that any extension of A has a com-pletely positive contractive section.

In the equivariant case, the same argument yields KKG� .A;B/ Š EG� .A;B/ if Ais nuclear and G acts properly on A (see also [55]). It should be possible to weakenproperness to amenability here, but I am not aware of a reference for this.

5 The Baum–Connes assembly map for spaces and operatoralgebras

The Baum–Connes conjecture is a guess for the K-theory K�.C�redG/ of reduced groupC�-algebras [4]. We shall compare the approach of Davis and Lück [14] using homotopytheory for G-spaces and its counterpart in bivariant K-theory formulated in [38]. Toavoid technical difficulties, we assume that the group G is discrete.

The first step in the Davis–Lück approach is to embed the groups of interest suchas K�.C�redG/ in a G-homology theory, that is, a homology theory on the categoryof (spectra of) G-CW-complexes. For the Baum–Connes assembly map we need ahomology theory for G-CW-complexes with F�.G=H/ Š K�.C�redH/. This amountsto finding a G-equivariant spectrum with appropriate homotopy groups [14] and is themost difficult part of the construction. Other interesting invariants like the algebraicK- and L-theory of group rings can be treated using other spectra instead.

32 R. Meyer

In the world of C�-algebras, we cannot treat algebraic K- and L-theory; but wehave much better tools to study K�.C�redG/. We do not need a G-homology theory buta homological functor on the triangulated category KKG . More precisely, we need ahomological functor that takes the value K�.C�redH/ on C0.G=H/ for all subgroupsH .The functor A 7! K�.G Ër A/ works fine here because G Ër C0.G=H;A/ is Morita–Rieffel equivalent to H Ër A for any H -C�-algebra A. The corresponding assertionfor full crossed products is known as Green’s Imprimitivity Theorem; reduced crossedproducts can be handled similarly. Thus a topological approach to the Baum–Connesconjecture forces us to consider K�.G Ër A/ for all G-C�-algebras A, which leads tothe Baum–Connes conjecture with coefficients.

5.1 Assembly maps via homotopy theory. Recall that a homology theory on pointedCW-complexes is determined by its value on S0. Similarly, aG-homology theory F isdetermined by its values F�.G=H/ on homogeneous spaces for all subgroupsH � G.This does not help much because these groups – which are K�.C�redH/ in the case ofinterest – are very hard to compute.

The idea behind assembly maps is to approximate a given homology theory bya simpler one that only depends on F�.G=H/ for H 2 F for some family of sub-groups F . The Baum–Connes assembly map uses the family of finite subgroups here;other families like virtually cyclic subgroups appear in isomorphism conjectures forother homology theories. We now fix a family of subgroups F , which we assume tobe closed under conjugation and subgroups.

A G;F -CW-complex is a G-CW-complex in which the stabilisers of cells belongto F . The universal G;F -CW-complex is a G;F -CW-complex E.G;F / with theproperty that, for any G;F -CW-complex X there is a G-map X ! E.G;F /, whichis unique up toG-homotopy. This universal property determines E.G;F / uniquely upto G-homotopy. It is easy to see that E.G;F / is H -equivariantly contractible for anyH 2 F . Conversely, a G;F -CW-complex with this property is universal.

Example 66. LetG D Z and let F be the family consisting only of the trivial subgroup;this agrees with the family of finite subgroups because G is torsion-free. A G;F -CW-complex is essentially the same as a CW-complex with a free cellular action of Z. It iseasy to check that R with the action of Z by translation and the usual cell decompositionis a universal G;F -CW-complex.

Given any G-CW-complex X , the canonical map E.G;F / � X ! X has thefollowing properties:

• E.G;F / �X is a G;F -CW-complex;

• if Y is a G;F -CW-complex, then any G-map Y ! X lifts uniquely up toG-homotopy to a map Y ! E.G;F / �X ;

• for any H 2 F , the map E.G;F / �X ! X becomes a homotopy equivalencein the category of H -spaces.


The first two properties make precise in what sense E.G;F / � X is the best approxi-mation to X among G;F -CW-complexes.

Definition 67. The assembly map with respect to F is the mapF��E.G;F /

�! F�.?/induced by the constant map E.G;F / D E.G;F / � ?! ?.

More generally, the assembly map with coefficients in a pointed G-CW-complex(or spectrum) X is the map F�.E.G;F /C ^X/! F�.S0 ^X/ D F�.X/ induced bythe map E.G;F /C ! ?C D S0.

In the stable homotopy category of pointed G-CW-complexes (or spectra), we getan exact triangle E.G;F /C ^ X ! X ! N ! S1 ^ E.G;F /C ^ X , where Nis H -equivariantly contractible for each H 2 F . This means that the domain of theassembly map F�.E.G;F /C ^ X/ is the localisation of F� at the class of all objectsthat are H -equivariantly contractible for each H 2 F .

Thus the assembly map is an isomorphism for all X if and only if F�.N / D 0

whenever N is H -equivariantly contractible for each H 2 F . Thus an isomorphismconjecture can be interpreted in two equivalent ways. First, it says that we can recon-struct the homology theory from its restriction to G;F -CW-complexes. Secondly, itsays that the homology theory vanishes for spaces that areH -equivariantly contractiblefor H 2 F .

5.2 From spaces to operator algebras. We can carry over the construction of as-sembly maps above to bivariant Kasparov theory; we continue to assume G discrete tosimplify some statements. From now on, we let F be the family of finite subgroups.This is the family that appears in the Baum–Connes assembly map. Other families ofsubgroups can also be treated, but some proofs have to be modified and are not yetwritten down.

First we need an analogue of G;F -CW-complexes. These are constructible out ofsimpler “cells” which we describe first, using the induction functors

IndGH W KKH ! KKG

for subgroups H � G. For a finite group H , IndGH .A/ is the H -fixed point algebra ofC0.G;A/, where H acts by h � f .g/ D ˛h

�f .gh/

�. For infinite H , we have

IndGH .A/ D ff 2 Cb.G;A/ j ˛hf .gh/ D f .g/ for all g 2 G, h 2 H ,

and gH 7! kf .g/k is in C0.G=H/gIthe group G acts by translations on the left.

This construction is functorial for equivariant �-homomorphisms. Since it com-mutes with C�-stabilisations and maps split extensions again to split extensions, itdescends to a functor KKH ! KKG by the universal property (compare §4.1).

We also have the more trivial restriction functors ResHG W KKG ! KKH for sub-groups H � G. The induction and restriction functors are adjoint:

KKG.IndGH A;B/ Š KKH .A;ResHG B/

34 R. Meyer

for all A 22 KKG ; this can be proved like the similar adjointness statements in §4.1,using the embedding A ! ResHG IndGH .A/ as functions supported on H � G andthe correspondence IndGH ResHG .A/ Š C0.G=H;A/ ! K.`2G=H/ ˝ A � A. It isimportant here thatH � G is an open subgroup. By the way, ifH � G is a cocompactsubgroup (which means finite index in the discrete case), then ResHG is the left-adjointof IndGH instead.

Definition 68. We let CI be the subcategory of all objects of KKG of the form IndGH .A/for A 22 KKH and H 2 F . Let hCIi be the smallest class in KKG that contains CI

and is closed under KKG-equivalence, countable direct sums, suspensions, and exacttriangles.

Equivalently, hCIi is the localising subcategory generated by CI. This is oursubstitute for the category of .G;F /-CW-complexes.

Definition 69. Let CC be the class of all objects of KKG with ResHG .A/ Š 0 for allH 2 F .

Theorem 70. If P 2 hCIi, N 2 CC , then KKG.P;N / D 0. Furthermore, for anyA 22 KKG there is an exact triangle P ! A! N ! †P with P 2 hCIi,N 2 CC .

Definitions 68–69 and Theorem 70 are taken from [38]. The map F�.P /! F�.A/for a functor F W KKG ! C is analogous to the assembly map in Definition 67 anddeserves to be called the Baum–Connes assembly map for F .

We can use the tensor product in KKG to simplify the proof of Theorem 70: oncewe have a triangle PC ! C ! NC ! †PC with PC 2 hCIi, NC 2 CC , then

A˝ PC ! A˝C ! A˝NC ! †A˝ PC

is an exact triangle with similar properties for A. It makes no difference whether weuse ˝min or ˝max here. The map PC ! C in KKG.PC;C/ is analogous to the mapE.G;F / ! ?. It is also called a Dirac morphism for G because the K-homologyclasses of Dirac operators on smooth spin manifolds provided the first important ex-amples [31].

The two assembly map constructions with spaces and C�-algebras are not justanalogous but provide the same Baum–Connes assembly map. To see this, we mustunderstand the passage from the homotopy category of spaces to KK. Usually, wemap spaces to operator algebras using the commutative C�-algebra C0.X/. But thisconstruction is only functorial for proper continuous maps, and the functoriality iscontravariant. The assembly map for, say, G D Z is related to the non-proper mapp W R! ?, which does not induce a map C ! C0.R/; even if it did, this map wouldstill go in the wrong direction. The wrong-way functoriality in KK provides an elementpŠ 2 KK1.C0.R/;C/ instead, which is the desired Dirac morphism up to a shift in thegrading. This construction only applies to manifolds with a Spinc-structure, but it canbe generalised as follows.

On the level of Kasparov theory, we can define another functor from suitable spacesto KK that is a covariant functor for all continuous maps. The definition uses a notion


of duality due to Kasparov [31] that is studied more systematically in [17]. It requiresyet another version RKKG� .X IA;B/ of Kasparov theory that is defined for a locallycompact space X and two G-C�-algebras A and B . Roughly speaking, the cycles forthis theory areG-equivariant families of cycles for KK�.A;B/ parametrised byX . Thegroups RKKG� .X IA;B/ are contravariantly functorial and homotopy invariant inX (forG-equivariant continuous maps).

We have RKKG� .?IA;B/ D KKG� .A;B/ and, more generally, RKKG� .X IA;B/ ŠKKG�

�A;C.X;B/

�if X is compact. The same statement holds for non-compact X ,

but the algebra C.X;B/ is not a C�-algebra any more: it is an inverse system ofC�-algebras.

Definition 71 ([17]). A G-C�-algebra PX is called an abstract dual for X if, forall second countable locally compact G-spaces Y and all separable G-C�-algebras Aand B , there are natural isomorphisms

RKKG.X � Y IA;B/ Š RKKG.Y IPX ˝ A;B/that are compatible with tensor products.

Abstract duals exist for many spaces. For trivial reasons, C is an abstract dualfor the one-point space. For a smooth manifold X with an isometric action of G, bothC0.T

�X/ and the algebra of C0-sections of the Clifford algebra bundle onX are abstractduals for X ; if X has a G-equivariant Spinc-structure – as in the example of Z actingon R – we may also use a suspension of C0.X/. For a finite-dimensional simplicialcomplex with a simplicial action of G, an abstract dual is constructed by GennadiKasparov and Georges Skandalis in [32] and in more detail in [17]. It seems likelythat the construction can be carried over to infinite-dimensional simplicial complexesas well, but this has not yet been written down.

There are also spaces with no abstract dual. A prominent example is the Cantor set:it has no abstract dual, even for trivial G (see [17]).

Let D be the class of allG-spaces that admit a dual. Recall thatX 7! RKKG.X �Y IA;B/ is a contravariant homotopy functor for continuousG-maps. Passing to corep-resenting objects, we get a covariant homotopy functor

D ! KKG ; X 7! PX :

This functor is very useful to translate constructions from homotopy theory to bivariantK-theory. An instance of this is the comparison of the Baum–Connes assembly mapsin both setups:

Theorem 72. Let F be the family of finite subgroups of a discrete group G, and letE.G;F / be the universal .G;F /-CW-complex. Then E.G;F / has an abstract dualP ,and the map E.G;F /! ? induces a Dirac morphism in KKG0 .P;C/.

Theorem 72 should hold for all families of subgroups F , but only the above specialcase is treated in [17], [38].

36 R. Meyer

5.3 The Dirac-dual-Dirac method and geometry. Let us compare the approachesin §5.1 and §5.2! The bad thing about the C�-algebraic approach is that it applies tofewer theories. The good thing about it is that Kasparov theory is so flexible that anycanonical map between K-theory groups has a fair chance to come from a morphismin KKG which we can construct explicitly.

For some groups, the Dirac morphism in KKG.P;C/ is a KK-equivalence:

Theorem73 (Higson–Kasparov [26]). Let the groupG be amenable or, more generally,a-T-menable. Then the Dirac morphism forG is a KKG-equivalence, so thatG satisfiesthe Baum–Connes conjecture with coefficients.

The class of groups for which the Dirac morphism has a one-sided inverse is evenlarger. This is the point of the Dirac-dual-Dirac method. The following definitionin [38] is based on a simplification of this method:

Definition 74. A dual Dirac morphism for G is an element 2 KKG.C; P / with ıD D idP .

If such a dual Dirac morphism exists, then it provides a section for the assemblymap F�.P ˝A/! F�.A/ for any functor F W KKG ! C and anyA 22 KKG , so thatthe assembly map is a split monomorphism. Currently, we know no group without adual Dirac morphism. It is shown in [16], [18], [19] that the existence of a dual Diracmorphism is a geometric property ofG because it is related to the invertibility of anotherassembly map that only depends on the coarse geometry ofG (in the torsion-free case).

Instead of going into this construction, we briefly indicate another point of view thatalso shows that the existence of a dual Dirac morphism is a geometric issue. Let P bean abstract dual for some spaceX (like E.G;F /). The duality isomorphisms in Defini-tion 71 are determined by two pieces of data: a Dirac morphismD 2 KKG.P;C/ anda local dual Dirac morphism ‚ 2 RKKG.X IC; P /. The notation is motivated by thespecial case of a Spinc-manifoldX withP D C0.X/, whereD is the K-homology classdefined by the Dirac operator and is defined by a local construction involving pointwiseClifford multiplications. If X D E.G;F /, then it turns out that 2 KKG.C; P / is adual Dirac morphism if and only if the canonical map KKG.C; P /! RKKG.X IC; P /maps 7! ‚. Thus the issue is to globalise the local construction of‚. This is possibleif we know, say, that X has non-positive curvature. This is essentially how Kasparovproves the Novikov conjecture for fundamental groups of non-positively curved smoothmanifolds in [31].

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Inheritance of isomorphism conjectures under colimits

Arthur Bartels, Siegfried Echterhoff, and Wolfgang Lück

1 Introduction

1.1 Assembly maps. We want to study the following assembly maps:

asmbGn W HGn .EVCyc.G/IKR/! HG

n .f�gIKR/ D Kn.R ÌG/I (1.1)

asmbGn W HGn .EF in.G/IKHR/! HG

n .f�gIKHR/ D KHn.R ÌG/I (1.2)

asmbGn W HGn .EVCyc.G/ILh�1iR /! HG

n .f�gILh�1iR / D Lh�1in .R ÌG/I (1.3)

asmbGn W HGn .EF in.G/IKtop

A;l1/! HG

n .f�gIKtopA;l1

/ D Kn.A Ìl1 G/I (1.4)


A;r/! HGn .f�gIKtop

A;r/ D Kn.A Ìr G/I (1.5)


A;m/! HGn .f�gIKtop

A;m/ D Kn.A Ìm G/: (1.6)

Some explanations are in order. A family of subgroups of G is a collection ofsubgroups of G which is closed under conjugation and taking subgroups. Examplesare the family F in of finite subgroups and the family VCyc of virtually cyclic sub-groups.

Let EF .G/ be the classifying space associated to F . It is uniquely characterizedup to G-homotopy by the properties that it is a G-CW-complex and that EF .G/

H iscontractible if H 2 F and is empty if H … F . For more information about thesespaces EF .G/ we refer for instance to the survey article [29].

Given a groupG acting on a ring (with involution) by structure preserving maps, letRÌG be the twisted group ring (with involution) and denote byKn.RÌG/, KHn.RÌG/andLh�1in .RÌG/ its algebraicK-theory in the non-connective sense (see Gersten [17]or Pedersen–Weibel [34]), its homotopy K-theory in the sense of Weibel [41], and itsL-theory with decoration �1 in the sense of Ranicki [37, Chapter 17]. Given a groupG acting on a C �-algebra A by automorphisms of C �-algebras, let A Ìl1 G be theBanach algebra obtained from A Ì G by completion with respect to the l1-norm, letA Ìr G be the reduced crossed product C �-algebra, and let A Ìm G be the maximalcrossed productC �-algebra and denote byKn.AÌl1G/,Kn.AÌrG/ andKn.AÌmG/their topological K-theory.

The source and target of the assembly maps are given by G-homology theories(see Definition 2.1 and Theorem 6.1) with the property that for every subgroupH � G

42 A. Bartels, S. Echterhoff, and W. Lück

and n 2 Z

HGn .G=H IKR/ Š Kn.R ÌH/I

HGn .G=H IKHR/ Š KHn.R ÌH/I

HGn .G=H ILh�1iR / Š Lh�1in .R ÌH/IHGn .G=H IKtop

A;l1/ Š Kn.A Ìl1 H/I

HGn .G=H IKtop

A;r/ Š Kn.A Ìr H/IHGn .G=H IKtop

A;r/ Š Kn.A Ìm H/:

All the assembly maps are induced by the projection from EF in.G/ or EVCyc.G/

respectively to the one-point-space f�g.Remark 1.7. It might be surprising to the reader that we restrict to C*-algebra co-efficients A in the assembly map (1.4). Indeed, our main results rely heavily on thevalidity of the conjecture for hyperbolic groups, which, so far, is only known for C*-algebra coefficients. Moreover we also want to study the passage from the l1-setting tothe C �-setting. Hence we decided to restrict ourselves to the case of C �-coefficientsthroughout. We mention that on the other hand the assembly map (1.4) can also bedefined for Banach algebra coefficients [33].

1.2 Conventions. Before we go on, let us fix some conventions. A group G is al-ways discrete. Hyperbolic group is to be understood in the sense of Gromov (see forinstance [11], [12], [18], [19]). Ring means associative ring with unit and ring ho-momorphisms preserve units. Homomorphisms of Banach algebras are assumed to benorm decreasing.

1.3 Isomorphism conjectures. The Farrell–Jones conjecture for algebraicK-theoryfor a groupG and a ringRwithG-action by ring automorphisms says that the assemblymap (1.1) is bijective for all n 2 Z. Its version for homotopy K-theory says that theassembly map (1.2) is bijective for all n 2 Z. If R is a ring with involution and Gacts on R by automorphism of rings with involutions, the L-theoretic version of theFarrell–Jones conjecture predicts that the assembly map (1.3) is bijective for all n 2 Z.The Farrell–Jones conjecture for algebraicK- and L-theory was originally formulatedin the paper by Farrell–Jones [15, 1.6 on page 257] for the trivial G-action on R. Itshomotopy K-theoretic version can be found in [4, Conjecture 7.3], again for trivialG-action on R.

Let G be a group acting on the C �-algebra A by automorphisms of C �-algebras.The Bost conjecture with coefficients and the Baum–Connes conjecture with coefficientsrespectively predict that the assembly map (1.4) and (1.5) respectively are bijectivefor all n 2 Z. The original statement of the Baum–Connes conjecture with trivialcoefficients can be found in [9, Conjecture 3.15 on page 254].

Inheritance of isomorphism conjectures under colimits 43

Our formulation of these conjectures follows the homotopy theoretic approach in[13]. The original assembly maps are defined differently. We do not give the proofthat our maps agree with the original ones but at least refer to [13, page 239], wherethe Farrell–Jones conjecture is treated and to Hambleton–Pedersen [21], where suchidentification is given for the Baum–Connes conjecture with trivial coefficients.

1.4 Inheritance under colimits. The main purpose of this paper is to prove thatthese conjectures are inherited under colimits over directed systems of groups (with notnecessarily injective structure maps). We want to show:

Theorem 1.8 (Inheritance under colimits). Let fGi j i 2 I g be a directed systemof groups with (not necessarily injective) structure maps �i;j W Gi ! Gj . Let G Dcolimi2I Gi be its colimit with structure maps i W Gi ! G. Let R be a ring (withinvolution) and letA be aC �-algebra with structure preservingG-action. Given i 2 Iand a subgroup H � Gi , we let H act on R and A by restriction with the grouphomomorphism . i /jH W H ! G. Fix n 2 Z. Then:

(i) If the assembly map

asmbHn W HHn .EVCyc.H/IKR/! HH

n .f�gIKR/ D Kn.R ÌH/

of (1.1) is bijective for all n 2 Z, all i 2 I and all subgroups H � Gi , then forevery subgroup K � G of G the assembly map

asmbKn W HKn .EVCyc.K/IKR/! HK

n .f�gIKR/ D Kn.R ÌK/

of (1.1) is bijective for all n 2 Z.

The corresponding version is true for the assembly maps given in (1.2), (1.3),(1.4), and (1.6).

(ii) Suppose that all structure maps �i;j are injective and that the assembly map

asmbGin W HGin .EVCyc.Gi /IKR/! HGi

n .f�gIKR/ D Kn.R ÌGi /

of (1.1) is bijective for all n 2 Z and i 2 I . Then the assembly map

asmbGn W HGn .EVCyc.G/IKR/! HG

n .f�gIKR/ D Kn.R ÌG/

of (1.1) is bijective for all n 2 Z.

The corresponding statement is true for the assembly maps given in (1.2), (1.3),(1.4), (1.5), and (1.6).

Theorem 1.8 will follow from Theorem 4.5 and Lemma 6.2 as soon as we haveproved Theorem 6.1. Notice that the version (1.5) does not appear in assertion (i). Acounterexample will be discussed below. The (fibered) version of Theorem 1.8 (i) inthe case of algebraicK-theory andL-theory with coefficients in Z with trivialG-actionhas been proved by Farrell–Linnell [16, Theorem 7.1].


1.5 Colimits of hyperbolic groups. In [23, Section 7] Higson, Lafforgue and Skan-dalis construct counterexamples to the Baum–Connes conjecture with coefficients, actu-ally with a commutative C �-algebra as coefficients. They formulate precise propertiesfor a group G which imply that it does not satisfy the Baum–Connes conjecture withcoefficients. Gromov [20] describes the construction of such a group G as a colimitover a directed system of groups fGi j i 2 I g, where each Gi is hyperbolic.

This construction did raise the hope that these groups G may also be counterex-amples to the Baum–Connes conjecture with trivial coefficients. But – to the authors’knowledge – this has not been proved and no counterexample to the Baum–Connesconjecture with trivial coefficients is known.

Of course one may wonder whether such counterexamples to the Baum–Connesconjecture with coefficients or with trivial coefficients respectively may also be coun-terexamples to the Farrell–Jones conjecture or the Bost conjecture with coefficients orwith trivial coefficients respectively. However, this can be excluded by the followingresult.

Theorem 1.9. Let G be the colimit of the directed system fGi j i 2 I g of groups (withnot necessarily injective structure maps). Suppose that each Gi is hyperbolic. LetK � G be a subgroup. Then:

(i) The group K satisfies for every ring R on which K acts by ring automorphismsthe Farrell–Jones conjecture for algebraic K-theory with coefficients in R, i.e.,the assembly map (1.1) is bijective for all n 2 Z.

(ii) The group K satisfies for every ring R on which K acts by ring automorphismsthe Farrell–Jones conjecture for homotopy K-theory with coefficients in R, i.e.,the assembly map (1.2) is bijective for all n 2 Z.

(iii) The group K satisfies for every C �-algebra A on which K acts by C �-auto-morphisms the Bost conjecture with coefficients inA, i.e., the assembly map (1.4)is bijective for all n 2 Z.

Proof. IfG is the colimit of the directed system fGi j i 2 I g, then the subgroupK � Gis the colimit of the directed system f �1i .K/ j i 2 I g, where i W Gi ! G is thestructure map. Hence it suffices to prove Theorem 1.9 in the case G D K. This casefollows from Theorem 1.8 (i) as soon as one can show that the Farrell–Jones conjecturefor algebraic K-theory, the Farrell–Jones conjecture for homotopy K-theory, or theBost conjecture respectively holds for every subgroupH of a hyperbolic groupG witharbitrary coefficients R and A respectively.

Firstly we prove this for the Bost conjecture. Mineyev and Yu [30, Theorem 17]show that every hyperbolic group G admits a G-invariant metric Od which is weaklygeodesic and strongly bolic. Since every subgroup H of G clearly acts properly on Gwith respect to any discrete metric, it follows thatH belongs to the class C 0 as describedby Lafforgue in [27, page 5] (see also the remarks at the top of page 6 in [27]). Nowthe claim is a direct consequence of [27, Theorem 0.0.2].


The claim for the Farrell–Jones conjecture is proved for algebraic K-theory andhomotopyK-theory in Bartels–Lück–Reich [6] which is based on the results of [5].

There are further groups with unusual properties that can be obtained as colimits ofhyperbolic groups. This class contains for instance a torsion-free non-cyclic group allwhose proper subgroups are cyclic constructed by Ol’shanskii [32]. Further examplesare mentioned in [31, p.5] and [38, Section 4].

We mention that if one can prove the L-theoretic version of the Farrell–Jones con-jecture for subgroups of hyperbolic groups with arbitrary coefficients, then it is alsotrue for subgroups of colimits of hyperbolic groups by the argument above.

1.6 Discussion of (potential) counterexamples. IfG is an infinite group which satis-fies Kazhdan’s property (T), then the assembly map (1.6) for the maximal group C �-algebra fails to be an isomorphism if the assembly map (1.5) for the reduced groupC �-algebra is injective (which is true for a very large class of groups and in particularfor all hyperbolic groups by [25]). The reason is that a group has property (T) ifand only if the trivial representation 1G is isolated in the dual yG of G. This impliesthat C �m.G/ has a splitting C ˚ ker.1G/, where we regard 1G as a representationof C �m.G/. If G is infinite, then the first summand is in the kernel of the regularrepresentation � W C �m.G/ ! C �r .G/ (see for instance [14]), hence the K-theory map� W K0.C �m.G//! K0.C

�r .G// is not injective. For a finite groupH we haveAÌrH D

A Ìm H and hence we can apply [13, Lemma 4.6] to identify the domains of (1.5)and (1.6). Under this identification the composition of the assembly map (1.6) with �is the assembly map (1.5) and the claim follows.

Hence the Baum–Connes conjecture for the maximal group C �-algebras is not truein general since the Baum–Connes conjecture for the reduced group C �-algebras hasbeen proved for some groups with property (T) by Lafforgue [26] (see also [39]). So inthe sequel our discussion refers always to the Baum–Connes conjecture for the reducedgroup C �-algebra.

One may speculate that the Baum–Connes conjecture with trivial coefficients isless likely to be true for a given group G than the Farrell–Jones conjecture or theBost conjecture. Some evidence for this speculation comes from lack of functorialityof the reduced group C �-algebra. A group homomorphism ˛ W H ! G induces ingeneral not a C �-homomorphism C �r .H/! C �r .G/, one has to require that its kernelis amenable. Here is a counterexample, namely, if F is a non-abelian free group, thenC �r .F / is a simple algebra [35] and hence there is no unital algebra homomorphismC �r .F /! C �r .f1g/ D C. On the other hand, any group homomorphism ˛ W H ! G

induces a homomorphism

HHn .EF in.H/IKtop

C;r/ind˛��! HG

n .˛�EF in.H/IKtopC;r/

HGn .f /��! HGn .EF in.G/IKtop

C;r/

whereG acts trivially on C and f W ˛�EF in.H/! EF in.G/ is the up toG-homotopyunique G-map. Notice that the induction map ind˛ exists since the isotropy groupsof EF in.H/ are finite. Moreover, this map is compatible under the assembly maps


for H and G with the map Kn.C �r .˛// W Kn.C �r .H// ! Kn.C�r .G// provided that

˛ has amenable kernel and hence C �r .˛/ is defined. So the Baum–Connes conjec-ture implies that every group homomorphism ˛ W H ! G induces a group homomor-phism˛� W Kn.C �r .H//! Kn.C

�r .G//, although there may be noC �-homomorphism

C �r .H/! C �r .G/ induced by˛. No such direct construction of˛� is known in general.

Here is another failure of the reduced groupC �-algebra. LetG be the colimit of thedirected system fGi j i 2 I g of groups (with not necessarily injective structure maps).Suppose that for every i 2 I and preimageH of a finite group under the canonical map i W Gi ! G the Baum–Connes conjecture for the maximal group C �-algebra holds(This is for instance true by [22] if ker. i / has the Haagerup property). Then

colimi2I HGin .EF in.Gi /IKtop

C;m/Š��! colimi2I HGi

n .E �

iF in.Gi /IKtop

C;m/

Š��! HGn .EF in.G/IKtop

C;m/

is a composition of two isomorphisms. The first map is bijective by the TransitivityPrinciple 4.3, the second by Lemma 3.4 and Lemma 6.2. This implies that the followingcomposition is an isomorphism

colimi2I HGin .EF in.Gi /IKtop

C;r/! colimi2I HGin .E �

iF in.Gi /IKtop

C;r/

! HGn .EF in.G/IKtop

C;r/

Namely, these two compositions are compatible with the passage from the maximal tothe reduced setting. This passage induces on the source and on the target isomorphismssince EF in.Gi / and EF in.G/ have finite isotropy groups, for a finite groupH we haveC �r .H/ D C �m.H/ and hence we can apply [13, Lemma 4.6]. Now assume furthermorethat the Baum–Connes conjecture for the reduced group C �-algebra holds for Gi foreach i 2 I and for G. Then we obtain an isomorphism

colimi2I Kn.C �r .Gi //Š��! Kn.C

�r .G//:

Again it is in general not at all clear whether there exists such a map in the case, wherethe structure maps i W Gi ! G do not have amenable kernels and hence do not inducemaps C �r .Gi /! C �r .G/.

These arguments do not apply to the Farrell–Jones conjecture or the Bost conjecture.Namely any group homomorphism ˛ W H ! G induces maps R Ì H ! R Ì G,A Ìl1 H ! A Ìl1 G, and A Ìm H ! A Ìm G for a ring R or a C �-algebra A withstructure preserving G-action, where we equip R and A with the H -action comingfrom ˛. Moreover we will show for a directed system fGi j i 2 I g of groups (withnot necessarily injective structure maps) andG D colimi2I Gi that there are canonical


isomorphisms (see Lemma 6.2)

colimi2I Kn.R ÌGi /Š��! Kn.R ÌG/I

colimi2I KHn.R ÌGi /Š��! KHn.R ÌG/I

colimi2I Lh�1in .R ÌGi /Š��! Lh�1in .R ÌG/I

colimi2I Kn.A Ìl1 Gi /Š��! Kn.A Ìl1 G/I

colimi2I Kn.A Ìm Gi /Š��! Kn.A Ìm G/:

Let A be a C �-algebra with G-action by C �-automorphisms. We can consider Aas a ring only. Notice that we get a commutative diagram

HGn .EVCyc.G/IKA/

��

�� KHn.A ÌG/

��

HGn .EVCyc.G/IKHA/ �� KHn.A ÌG/

HGn .EF in.G/IKHA/

��

Š��

�� KHn.A ÌG/

��

id

��

HGn .EF in.G/IKtop

A;l1/

Š��

�� Kn.A Ìl1 G/

��

HGn .EF in.G/IKtop

A;m/

Š��

�� Kn.A Ìm G/

��

HGn .EF in.G/IKtop

A;r/�� Kn.A Ìr G/,

where the horizontal maps are assembly maps and the vertical maps are change oftheory and rings maps or induced by the up toG-homotopy uniqueG-mapEF in.G/!EVCyc.G/. The second left vertical map, which is marked with Š, is bijective. Thisis shown in [4, Remark 7.4] in the case, where G acts trivially on R, the proof carriesdirectly over to the general case. The fourth and fifth vertical left arrow, which aremarked with Š, are bijective, since for a finite group H we have A Ì H D A Ìl1H D A Ìr H D A Ìm H and hence we can apply [13, Lemma 4.6]. In particularthe Bost conjecture and the Baum–Connes conjecture together imply that the mapKn.A Ìl1 G/! Kn.A Ìr G/ is bijective, the map Kn.A Ìl1 G/! Kn.A Ìm G/ issplit injective and the map Kn.A Ìm G/! Kn.A Ìr G/ is split surjective.

The upshot of this discussions is:


• The counterexamples of Higson, Lafforgue and Skandalis [23, Section 7] tothe Baum–Connes conjecture with coefficients are not counterexamples to theFarrell–Jones conjecture or the Bost conjecture.

• The counterexamples of Higson, Lafforgue and Skandalis [23, Section 7] showthat the map Kn.A Ìl1 G/! Kn.A Ìr G/ is in general not bijective.

• The passage from the topological K-theory of the Banach algebra l1.G/ to thereduced group C �-algebra is problematic and may cause failures of the Baum–Connes conjecture.

• The Bost conjecture and the Farrell–Jones conjecture are more likely to be truethan the Baum–Connes conjecture.

• There is – to the authors’ knowledge – no promising candidate of a group G forwhich a strategy is in sight to show that the Farrell–Jones conjecture or the Bostconjecture are false. (Whether it is reasonable to believe that these conjecturesare true for all groups is a different question.)

1.7 Homology theories and spectra. The general strategy of this paper is to presentmost of the arguments in terms of equivariant homology theories. Many of the ar-guments for the Farrell–Jones conjecture, the Bost conjecture or the Baum–Connesconjecture become the same, the only difference lies in the homology theory we applythem to. This is convenient for a reader who is not so familiar with spectra and prefersto think of K-groups and not of K-spectra.

The construction of these equivariant homology theories is a second step and donein terms of spectra. Spectra cannot be avoided in algebraic K-theory by definitionand since we want to compare also algebraic and topological K-theory, we need spec-tra descriptions here as well. Another nice feature of the approach to equivarianttopological K-theory via spectra is that it yields a theory which can be applied toall G-CW-complexes. This will allow us to consider in the case G D colimi2I Gi theequivariantK-homology of theGi -CW-complex �i EF in.G/ D E �F in.Gi / although �i EF in.G/ has infinite isotropy groups if the structure map i W Gi ! G has infinitekernel.

Details of the constructions of the relevant spectra, namely, the proof of Theorem 8.1,will be deferred to [2]. We will use the existence of these spectra as a black box. Theseconstructions require some work and technical skills, but their details are not at allrelevant for the results and ideas of this paper and their existence is not at all surprising.

1.8 Twisting by cocycles. In the L-theory case one encounters also non-orientablemanifolds. In this case twisting with the first Stiefel–Whitney class is required. Ina more general setup one is given a group G, a ring R with involution and a grouphomomorphism w W G ! cent.R�/ to the center of the multiplicative group of unitsin R. So far we have used the standard involution on the group ring RG, which isgiven by r � g D r � g�1. One may also consider the w-twisted involution given by


r � g D rw.g/ � g�1. All the results in this paper generalize directly to this casesince one can construct a modified L-theory spectrum functor (over G) using the w-twisted involution and then the homology arguments are just applied to the equivarianthomology theory associated to this w-twisted L-theory spectrum.

Acknowledgements. The work was financially supported by the Sonderforschungs-bereich 478 – Geometrische Strukturen in der Mathematik – and the Max-Planck-Forschungspreis of the third author.

2 Equivariant homology theories

In this section we briefly explain basic axioms, notions and facts about equivarianthomology theories as needed for the purposes of this article. The main examples whichwill play a role in connection with the Bost, the Baum–Connes and the Farrell–Jonesconjecture will be presented later in Theorem 6.1.

Fix a group G and a ring ƒ. In most cases ƒ will be Z. The following definitionis taken from [28, Section 1].

Definition 2.1 (G-homology theory). A G-homology theory HG� with values in ƒ-modules is a collection of covariant functors HG

n from the category of G-CW-pairs tothe category of ƒ-modules indexed by n 2 Z together with natural transformations@Gn .X;A/ W HG

n .X;A/! HGn�1.A/ WD HG

n�1.A;;/ for n 2 Z such that the followingaxioms are satisfied:

• G-homotopy invariance. If f0 and f1 are G-homotopic maps .X;A/! .Y; B/

of G-CW-pairs, then HGn .f0/ D HG

n .f1/ for n 2 Z.

• Long exact sequence of a pair. Given a pair .X;A/ of G-CW-complexes, thereis a long exact sequence

� � �HGnC1

.j /

��! HGnC1.X;A/

@GnC1��! HG

n .A/

HGn .i/��! HG

n .X/HGn .j /��! HG

n .X;A/@Gn��! � � � ;

where i W A! X and j W X ! .X;A/ are the inclusions.

• Excision. Let .X;A/ be a G-CW-pair and let f W A ! B be a cellular G-mapof G-CW-complexes. Equip .X [f B;B/ with the induced structure of a G-CW-pair. Then the canonical map .F; f / W .X;A/ ! .X [f B;B/ induces anisomorphism

HGn .F; f / W HG

n .X;A/Š��! HG

n .X [f B;B/:


• Disjoint union axiom. Let fXi j i 2 I g be a family ofG-CW-complexes. Denoteby ji W Xi !`

i2I Xi the canonical inclusion. Then the map

Mi2I

HGn .ji / W

Mi2I

HGn .Xi /

Š��! HGn

� ai2I

Xi

�

is bijective.

Let HG� and KG� be G-homology theories. A natural transformation T� W HG� !KG� ofG-homology theories is a sequence of natural transformations Tn W HG

n !KGn

of functors from the category of G-CW-pairs to the category of ƒ-modules which arecompatible with the boundary homomorphisms.

Lemma 2.2. LetT� W HG� !KG� be a natural transformation ofG-homology theories.Suppose that Tn.G=H/ is bijective for every homogeneous space G=H and n 2 Z.

Then Tn.X;A/ is bijective for every G-CW-pair .X;A/ and n 2 Z.

Proof. The disjoint union axiom implies that bothG-homology theories are compatiblewith colimits over directed systems indexed by the natural numbers (such as the systemgiven by the skeletal filtration X0 � X1 � X2 � � � � � [n�0Xn D X ). The argumentfor this claim is analogous to the one in [40, 7.53]. Hence it suffices to prove thebijectivity for finite-dimensional pairs. Using the axioms of a G-homology theory, thefive lemma and induction over the dimension one reduces the proof to the special case.X;A/ D .G=H;;/.

Next we present a slight variation of the notion of an equivariant homology theoryintroduced in [28, Section 1]. We have to treat this variation since we later want to studycoefficients over a fixed group� which we will then pullback via group homomorphismswith � as target. Namely, fix a group � . A group .G; �/ over � is a group G togetherwith a group homomorphism � W G ! � . A map ˛ W .G1; �1/ ! .G2; �2/ of groupsover � is a group homomorphisms ˛ W G1 ! G2 satisfying �2 ı ˛ D �1.

Let ˛ W H ! G be a group homomorphism. Given an H -space X , define theinduction ofX with ˛ to be theG-space denoted by ˛�X which is the quotient ofG�Xby the right H -action .g; x/ � h WD .g˛.h/; h�1x/ for h 2 H and .g; x/ 2 G � X .If ˛ W H ! G is an inclusion, we also write indGH instead of ˛�. If .X;A/ is anH -CW-pair, then ˛�.X;A/ is a G-CW-pair.

Definition 2.3 (Equivariant homology theory over a group �). An equivariant homol-ogy theory H ‹� with values inƒ-modules over a group � assigns to every group .G; �/over � a G-homology theory HG� with values in ƒ-modules and comes with the fol-lowing so called induction structure: given a homomorphism ˛ W .H; �/ ! .G;�/ ofgroups over � and an H -CW-pair .X;A/, there are for each n 2 Z natural homomor-phisms

ind˛ W HHn .X;A/! HG

n .˛�.X;A// (2.4)

satisfying


• Compatibility with the boundary homomorphisms. @Gn ı ind˛ D ind˛ ı@Hn .

• Functoriality. Let ˇ W .G;�/! .K; �/ be another morphism of groups over � .Then we have for n 2 Z

indˇı˛ D HKn .f1/ ı indˇ ı ind˛ W HH

n .X;A/! HKn ..ˇ ı ˛/�.X;A//;

where f1 W ˇ�˛�.X;A/ Š�! .ˇ ı˛/�.X;A/, .k; g; x/ 7! .kˇ.g/; x/ is the naturalK-homeomorphism.

• Compatibility with conjugation. Let .G; �/ be a group over� and let g 2 G be anelement with �.g/ D 1. Then the conjugation homomorphisms c.g/ W G ! G

defines a morphism c.g/ W .G; �/! .G; �/ of groups over � . Let f2 W .X;A/!c.g/�.X;A/ be the G-homeomorphism which sends x to .1; g�1x/ in G �c.g/.X;A/.

Then for every n 2 Z and every G-CW-pair .X;A/ the homomorphism

indc.g/ W HGn .X;A/! HG

n .c.g/�.X;A//

agrees with HGn .f2/.

• Bijectivity. If ˛ W .H; �/ ! .G;�/ is a morphism of groups over � such thatthe underlying group homomorphism ˛ W H ! G is an inclusion of groups, thenind˛ W HH

n .f�g/! HGn .˛�f�g/ D HG

n .G=H/ is bijective for all n 2 Z.

Definition 2.3 reduces to the one of an equivariant homology in [28, Section 1] ifone puts � D f1g.Lemma 2.5. Let ˛ W .H; �/! .G;�/ be a morphism of groups over � . Let .X;A/ bean H -CW-pair such that ker.˛/ acts freely on X � A. Then


n .˛�.X;A//

is bijective for all n 2 Z.

Proof. Let F be the set of all subgroups ofH whose intersection with ker.˛/ is trivial.Obviously, this is a family, i.e., closed under conjugation and taking subgroups. A H -CW-pair .X;A/ is called a F -H -CW-pair if the isotropy group of any point in X �Abelongs to F . AH -CW-pair .X;A/ is a F -H -CW-pair if and only if ker.˛/ acts freelyon X � A.

The n-skeleton of ˛�.X;A/ is ˛� applied to the n-skeleton of .X;A/. Let .X;A/ bean H -CW-pair and let f W A ! B be a cellular H -map of H -CW-complexes. Equip.X [f B;B/ with the induced structure of a H -CW-pair. Then there is an obviousnatural isomorphism of G-CW-pairs

˛�.X [f B;B/ Š��! .˛�X [˛�f ˛�B; ˛�B/:


Now we proceed as in the proof of Lemma 2.2 but now considering the transfor-mations


n .˛�.X;A//

only for F -H -CW-pairs .X;A/. Thus we can reduce the claim to the special case.X;A/ D H=L for some subgroup L � H with L \ ker.˛/ D f1g. This specialcase follows from the following commutative diagram whose vertical arrows are bijec-tive by the axioms and whose upper horizontal arrow is bijective since ˛ induces anisomorphism ˛jL W L! ˛.L/:

HLn .f�g/

ind˛jL W L!˛.L/��

indHL

��

H˛.L/n .f�g/

indG˛.L/

��

HHn .H=L/

ind˛ �� HGn .˛�H=L/ D HG

n .G=˛.L//.

3 Equivariant homology theories and colimits

Fix a group� and an equivariant homology theory H ‹�with values inƒ-modules over� .Let X be a G-CW-complex. Let ˛ W H ! G be a group homomorphism. Denote

by ˛�X the H -CW-complex obtained from X by restriction with ˛. We have alreadyintroduced the induction ˛�Y of an H -CW-complex Y . The functors ˛� and ˛� areadjoint to one another. In particular the adjoint of the identity on ˛�X is a naturalG-map

f .X; ˛/ W ˛�˛�X ! X: (3.1)

It sends an element in G �˛ ˛�X given by .g; x/ to gx.Consider a map ˛ W .H; �/! .G;�/ of groups over � . Define the ƒ-map

an D an.X; ˛/ W HHn .˛

�X/ ind˛��! HGn .˛�˛�X/

HGn .f .X;˛//��! HG

n .X/:

If ˇ W .G;�/ ! .K; �/ is another morphism of groups over � , then by the axioms of

an induction structure the composite HHn .˛

�ˇ�X/an.ˇ

�X;˛/��! HGn .ˇ

�X/an.X;ˇ/��!

HKn .X/ agrees with an.X; ˇ ı ˛/ W HH

n .˛�ˇ�X/ D HH

n ..ˇ ı ˛/�X/! HGn .X/ for

a K-CW-complex X .Consider a directed system of groups fGi j i 2 I g with G D colimi2I Gi and

structure maps i W Gi ! G for i 2 I and �i;j W Gi ! Gj for i; j 2 I; i � j . Weobtain for every G-CW-complex X a system of ƒ-modules fHGi . �i X/ j i 2 I gwith structure maps an. �j X; �i;j / W HGi . �i X/ ! HGj . �j X/. We get a map ofƒ-modules

tGn .X;A/ WD colimi2I an.X; i / W colimi2I HGin . �i .X;A// ! HG

n .X;A/:(3.2)

The next definition is an extension of [4, Definition 3.1].


Definition 3.3 ((Strongly) continuous equivariant homology theory). An equivarianthomology theory H ‹� over � is called continuous if for every group .G; �/ over � andevery directed system of subgroups fGi j i 2 I g of G with G D S

i2I Gi the ƒ-map(see (3.2))

tGn .f�g/ W colimi2I HGin .f�g/! HG

n .f�g/is an isomorphism for every n 2 Z.

An equivariant homology theory H ‹� over � is called strongly continuous if forevery group .G; �/ over � and every directed system of groups fGi j i 2 I g withG D colimi2I Gi and structure maps i W Gi ! G for i 2 I the ƒ-map

tGn .f�g/ W colimi2I HGin .f�g/! HG

n .f�g/

is an isomorphism for every n 2 Z.

Here and in the sequel we view Gi as a group over � by � ı i W Gi ! � and i W Gi ! G as a morphism of groups over � .

Lemma 3.4. Let .G; �/ be a group over � . Consider a directed system of groupsfGi j i 2 I g with G D colimi2I Gi and structure maps i W Gi ! G for i 2 I . Let.X;A/ be a G-CW-pair. Suppose that H ‹� is strongly continuous.

Then the ƒ-homomorphism (see (3.2))

tGn .X;A/ W colimi2I HGin . �i .X;A//

Š��! HGn .X;A/

is bijective for every n 2 Z.

Proof. The functor sending a directed systems of ƒ-modules to its colimit is an exactfunctor and compatible with direct sums over arbitrary index maps. If .X;A/ is a pairof G-CW-complexes, then . �i X; �i A/ is a pair of Gi -CW-complexes. Hence thecollection of maps ftGn .X;A/ j n 2 Zg is a natural transformation of G-homologytheories of pairs ofG-CW-complexes which satisfy the disjoint union axiom. Hence inorder to show that tGn .X;A/ is bijective for all n 2 Z and all pairs ofG-CW-complexes.X;A/, it suffices by Lemma 2.2 to prove this in the special case .X;A/ D .G=H;;/.

For i 2 I let ki W Gi= �1i .H/ ! �i .G=H/ be the Gi -map sending gi �1i .H/

to i .gi /H . Consider a directed system of ƒ-modules fHGin .Gi=

�1i .H// j i 2 I g

whose structure maps for i; j 2 I; i � j are given by the composite

HGin .Gi=

�1i .H//

ind�i;j��! HGjn .Gj ��i;j Gi= �1i .H//

HGjn .fi;j /��! H

Gjn .Gj =

�1j .H//

for theGj -map fi;j W Gj ��i;j Gi= �1i .H/! Gj = �1j .H/ sending .gj ; gi �1i .H//


to gj�i;j .gi / �1j .H/. Then the following diagram commutes:

colimi2I H �1i.H/

n .f�g/colimi2I ind

Gi

�1i.H/

Š��

tHn .f�g/ Š

��

colimi2I HGin .Gi=

�1i .H//

colimi2I HGin .ki /

��

colimi2I HGin . �i .G=H//

tGn .G=H/

��

HHn .f�g/

indGH

Š�� HG

n .G=H/,

where the horizontal maps are the isomorphism given by induction. For the directedsystem f �1i .H/ j i 2 I g with structure maps �i;j j �1

i.H/ W �1i .H/ ! �1j .H/,

the group homomorphism colimi2I i j �1i.H/ W colimi2I �1i .H/! H is an isomor-

phism. This follows by inspecting the standard model for the colimit over a directedsystem of groups. Hence the left vertical arrow is bijective since H ‹� is strongly con-tinuous by assumption. Therefore it remains to show that the map

colimi2I HGin .ki / W colimi2I HGi

n .Gi= �1i .H//!colimi2I HGi

n . �i G=H/ (3.5)

is surjective.Notice that the map given by the direct sum of the structure maps

Mi2I

HGin . �i G=H/! colimi2I HGi

n . �i G=H/

is surjective. Hence it remains to show for a fixed i 2 I that the image of the structuremap

HGin . �i G=H/! colimi2I HGi

n . �i G=H/is contained in the image of the map (3.5).

We have the decomposition of the Gi -set �i G=H into its Gi -orbits

aGi .gH/2Gin. �

iG=H/

Gi= �1i .gHg�1/ Š��! �i G=H; gi

�1i .gHg�1/ 7! i .gi /gH:

It induces an identification of ƒ-modulesMGi .gH/2Gin. �

iG=H/

HGin

�Gi=

�1i .gHg�1/

� D HGin . �i G=H/:

Hence it remains to show for fixed elements i 2 I and Gi .gH/ 2 Gin. �i G=H/ thatthe obvious composition

HGin

�Gi=

�1i .gHg�1/

� � HGin . �i G=H/! colimi2I HGi

n . �i G=H/


is contained in the image of the map (3.5).Choose an index j with j � i and g 2 im. j /. Then the structure map for i � j is

a map HGin . �i G=H/! H

Gjn . �j G=H/ which sends the summand corresponding to

Gi .gH/ 2 Gin. �i G=H/ to the summand corresponding toGj .1H/ 2 Gj n. �j G=H/which is by definition the image of

HGjn .kj / W HGj

n .Gj = �1j .H//! H

Gjn . �j G=H/:

Obviously the image of composite of the last map with the structure map

HGjn . �j G=H/! colimi2I HGi

n . �i G=H/

is contained in the image of the map (3.5). Hence the map (3.5) is surjective. Thisfinishes the proof of Lemma 3.4.

4 Isomorphism conjectures and colimits

A family F of subgroups of G is a collection of subgroups of G which is closed underconjugation and taking subgroups. Let EF .G/ be the classifying space associated toF . It is uniquely characterized up to G-homotopy by the properties that it is a G-CW-complex and thatEF .G/

H is contractible ifH 2 F and is empty ifH … F . For moreinformation about these spaces EF .G/ we refer for instance to the survey article [29].Given a group homomorphism � W K ! G and a family F of subgroups of G, definethe family ��F of subgroups of K by

��F D fH � K j �.H/ 2 F g: (4.1)

If � is an inclusion of subgroups, we also write F jK instead of ��F .

Definition 4.2 (Isomorphism conjecture for H ‹�). Fix a group � and an equivarianthomology theory H ‹� with values in ƒ-modules over � .

A group .G; �/ over � together with a family of subgroups F of G satisfies theisomorphism conjecture (for H ‹�) if the projection pr W EF .G/! f�g to the one-point-space f�g induces an isomorphism

HGn .pr/ W HG

n .EF .G//Š��! HG

n .f�g/for all n 2 Z.

From now on fix a group � and an equivariant homology theory H ‹� over � .

Theorem 4.3 (Transitivity principle). Let .G; �/ be a group over � . Let F � G befamilies of subgroups of G. Assume that for every element H 2 G the group .H; �jH /over � satisfies the isomorphism conjecture for F jH .

Then the up to G-homotopy unique map EF .G/ ! EG .G/ induces an isomor-phism HG

n .EF .G//! HGn .EG .G// for all n 2 Z. In particular, .G; �/ satisfies the

isomorphism conjecture for G if and only if .G; �/ satisfies the isomorphism conjecturefor F .


Proof. The proof is completely analogous to the one in [4, Theorem 2.4, Lemma 2.2],where only the case � D f1g is treated.

Theorem 4.4. Let .G; �/ be a group over � . Let F be a family of subgroups of G.

(i) Let G be the directed union of subgroups fGi j i 2 I g. Suppose that H ‹� iscontinuous and for every i 2 I the isomorphism conjecture holds for .Gi ; �jGi /and F jGi .Then the isomorphism conjecture holds for .G; �/ and F .

(ii) Let fGi j i 2 I g be a directed system of groups with G D colimi2I Gi andstructure maps i W Gi ! G. Suppose that H ‹� is strongly continuous and forevery i 2 I the isomorphism conjecture holds for .Gi ; � ı i / and �i F .

Then the isomorphism conjecture holds for .G; �/ and F .

Proof. (i) The proof is analogous to the one in [4, Proposition 3.4].(ii) This follows from the following commutative square whose horizontal arrows

are bijective because of Lemma 3.4 and the identification �i EF .G/ D E �

iF .Gi /:

colimi2I HGin .E �

iF .Gi //

tGn .EF .G//

Š��

��

HGn .EF .G//

��

colimi2I HGin .f�g/ tGn .f�g/

Š�� HG

n .f�g/.

Fix a class of groups C closed under isomorphisms, taking subgroups and takingquotients, e.g., the class of finite groups or the class of virtually cyclic groups. For agroup G let C.G/ be the family of subgroups of G which belong to C .

Theorem 4.5. Let .G; �/ be a group over � .

(i) Let G be the directed union G D Si2I Gi of subgroups Gi Suppose that H ‹� is

continuous and that the isomorphism conjecture is true for .Gi ; �jGi / and C.Gi /

for all i 2 I .

Then the isomorphism conjecture is true for .G; �/ and C.G/.

(ii) Let fGi j i 2 I g be a directed system of groups with G D colimi2I Gi andstructure maps i W Gi ! G. Suppose that H ‹� is strongly continuous and thatthe isomorphism conjecture is true for .H;C.H// for every i 2 I and everysubgroup H � Gi .Then for every subgroupK � G the isomorphism conjecture is true for .K; �jK/and C.K/.


Proof. (i) This follows from Theorem 4.4 (i) since C.Gi / D C.G/jGi holds for i 2 I .(ii) IfG is the colimit of the directed system fGi j i 2 I g, then the subgroupK � G

is the colimit of the directed system f �1i .K/ j i 2 I g. Hence we can assume G D Kwithout loss of generality.

Since C is closed under quotients by assumption, we have C.Gi / � �i C.G/ forevery i 2 I . Hence we can consider for any i 2 I the composition

HGin .EC.Gi /.Gi //! HGi

n .E �

iC.G/.Gi //! HGi

n .f�g/:

Because of Theorem 4.4 (ii) it suffices to show that the second map is bijective. Byassumption the composition of the two maps is bijective. Hence it remains to show thatthe first map is bijective. By Theorem 4.3 this follows from the assumption that theisomorphism conjecture holds for every subgroup H � Gi and in particular for anyH 2 �i C.G/ for C.Gi /jH D C.H/.

5 Fibered isomorphism conjectures and colimits

In this section we also deal with the fibered version of the isomorphism conjectures.(This is not directly needed for the purpose of this paper and the reader may skip thissection.) This is a stronger version of the Farrell–Jones conjecture. The Fibered Farrell–Jones conjecture does imply the Farrell–Jones conjecture and has better inheritanceproperties than the Farrell–Jones conjecture.

We generalize (and shorten the proof of) the result of Farrell–Linnell [16, Theo-rem 7.1] to a more general setting about equivariant homology theories as developedin Bartels–Lück [3].

Definition 5.1 (Fibered isomorphism conjecture for H ‹�). Fix a group � and an equiv-ariant homology theory H ‹� with values in ƒ-modules over � . A group .G; �/ over �together with a family of subgroups F ofG satisfies the fibered isomorphism conjecture(for H ‹�) if for each group homomorphism � W K ! G the group .K; � ı �/ over �satisfies the isomorphism conjecture with respect to the family ��F .

Theorem 5.2. Let .G; �/ be a group over � . Let F be a family of subgroups ofG. LetfGi j i 2 I g be a directed system of groups withG D colimi2I Gi and structure maps i W Gi ! G. Suppose that H ‹� is strongly continuous and for every i 2 I the fiberedisomorphism conjecture holds for .Gi ; � ı i / and �i F .

Then the fibered isomorphism conjecture holds for .G; �/ and F .

Proof. Let � W K ! G be a group homomorphism. Consider the pullback of groups

Ki

S i��

�i �� Gi

i

��

K�

�� G.


Explicitly Ki D f.k; gi / 2 K � Gi j �.k/ D i .gi /g. Let x�i;j W Ki ! Kj be themap induced by �i;j W Gi ! Gj , idK and idG and the pullback property. One easilychecks by inspecting the standard model for the colimit over a directed set that weobtain a directed system x�i;j W Ki ! Kj of groups indexed by the directed set I and

the system of maps x i W Ki ! K yields an isomorphism colimi2I KiŠ�! K. The

following diagram commutes:

colimi2I HKin

�S i��EF .G/� tKn .�

�EF .G//

Š��

��

HKn .�

�EF .G//

��

colimi2I HKin .f�g/ tKn .f�g/

Š�� HK

n .f�g/,

where the vertical arrows are induced by the obvious projections onto f�g and thehorizontal maps are the isomorphisms from Lemma 3.4. Notice that S i��EF .G/ is amodel for ES i�

��F.Ki / D E��

i �

iF .Ki /. Hence each map H

Kin

�S i��EF .G/�!

HKin .f�g/ is bijective since .Gi ; � ı i / satisfies the fibered isomorphism conjecture for

�i F and hence .Ki ; � ı i ı �i / satisfies the isomorphism conjecture for ��i �i F .This implies that the left vertical arrow is bijective. Hence the right vertical arrow isan isomorphism. Since ��EF .G/ is a model for E��F .K/, this means that .K; � ı�/ satisfies the isomorphism conjecture for ��F . Since � W K ! G is any grouphomomorphism, .G; �/ satisfies the fibered isomorphism conjecture for F .

The proof of the following results are analogous to the one in [3, Lemma 1.6] and [4,Lemma 1.2], where only the case � D f1g is treated.

Lemma 5.3. Let .G; �/ be a group over � and let F � G be families of subgroupsofG. Suppose that .G; �/ satisfies the fibered isomorphism conjecture for the family F .

Then .G; �/ satisfies the fibered isomorphism conjecture for the family G .

Lemma 5.4. Let .G; �/ be a group over � . Let � W K ! G be a group homomorphismand let F be a family of subgroups of G. If .G; �/ satisfies the fibered isomorphismconjecture for the family F , then .K; � ı�/ satisfies the fibered isomorphism conjecturefor the family ��F .

For the remainder of this section fix a class of groups C closed under isomorphisms,taking subgroups and taking quotients, e.g., the families F in or VCyc.

Lemma 5.5. Let .G; �/ be a group over � . Suppose that the fibered isomorphismconjecture holds for .G; �/ and C.G/. Let H � G be a subgroup.

Then the fibered isomorphism conjecture holds for .H; �jH / and C.H/.

Proof. This follows from Lemma 5.4 applied to the inclusion H ! G since C.H/ DC.G/jH .

Theorem 5.6. Let .G; �/ be a group over � .


(i) Let G be the directed union G D Si2I Gi of subgroups Gi . Suppose that H ‹�

is continuous and that the fibered isomorphism conjecture is true for .Gi ; �jGi /and C.Gi / for all i 2 I .

Then the fibered isomorphism conjecture is true for .G; �/ and C.G/.

(ii) Let fGi j i 2 I g be a directed system of groups with G D colimi2I Gi andstructure maps i W Gi ! G. Suppose that H ‹� is strongly continuous and thatthe fibered isomorphism conjecture is true for .Gi ; � ı i / and C.Gi // for alli 2 I .

Then the fibered isomorphism conjecture is true for .G; �/ and C.G/.

Proof. (i) The proof is analogous to the one in [4, Proposition 3.4], where the case� D f1g is considered.

(ii) Because C is closed under taking quotients we conclude C.Gi / � �i C.G/.Now the claim follows from Theorem 5.2 and Lemma 5.3.

Corollary 5.7. (i) Suppose that H ‹� is continuous. Then the ( fibered) isomorphismconjecture for .G; �/ and C.G/ is true for all groups .G; �/ over � if and only if it istrue for all such groups where G is a finitely generated group.

(ii) Suppose that H ‹� is strongly continuous. Then the fibered isomorphism conjec-ture for .G; �/ and C.G/ is true for all groups .G; �/ over � if and only if it is true forall such groups where G is finitely presented.

Proof. Let .G; �/ be a group over � where G is finitely generated. Choose a finitelygenerated free groupF together with an epimorphism W F ! G. LetK be the kernelof . Consider the directed system of finitely generated subgroups fKi j i 2 I g of K.Let SKi be the smallest normal subgroup of K containing Ki . Explicitly SKi is givenby elements which can be written as finite products of elements of the shape f kif �1for f 2 F and k 2 Ki . We obtain a directed system of groups fF=SKi j i 2 I g,where for i � j the structure map �i;j W F=SKi ! F= SKj is the canonical projection.If i W F=SKi ! F=K D G is the canonical projection, then the collection of maps

f i j i 2 I g induces an isomorphism colimi2I F=SKi Š�! G. By construction for eachi 2 I the group F=SKi is finitely presented and the fibered isomorphism conjectureholds for .F=SKi ; � ı i / and C.F=SKi / by assumption. Theorem 5.6 (ii) implies thatthe fibered Farrell–Jones conjecture for .G; �/ and C.G/ is true.

6 Some equivariant homology theories

In this section we will describe the relevant homology theories over a group � andshow that they are (strongly) continuous. (We have defined the notion of an equivarianthomology theory over a group in Definition 2.3.)


6.1 Desired equivariant homology theories. We will need the following

Theorem 6.1 (Construction of equivariant homology theories). Suppose that we aregiven a group � and a ring R (with involution) or a C �-algebra A respectively onwhich � acts by structure preserving automorphisms. Then:

(i) Associated to these data there are equivariant homology theories with values inZ-modules over the group �

H ‹�.�IKR/

H ‹�.�IKHR/

H ‹�.�ILh�1iR /;

H ‹�.�IKtopA;l1

/;

H ‹�.�IKtopA;r/;

H ‹�.�IKtopA;m/;

where in the case H ‹�.�IKtopA;r/ we will have to impose the restriction to the

induction structure that a homomorphisms ˛ W .H; �/! .G;�/ over � inducesa transformation ind˛ W HH

n .X;X0/! HGn .˛�.X;X0// only if the kernel of the

underlying group homomorphism ˛ W H ! G acts with amenable isotropy onX nX0.

(ii) If .G;�/ is a group over � and H � G is a subgroup, then there are for everyn 2 Z identifications

HHn .f�gIKR/ Š HG

n .G=H IKR/ Š Kn.R ÌH/IHHn .f�gIKHR/ Š HG

n .G=H IKHR/ Š KHn.R ÌH/IHHn .f�gILh�1iR / Š HG

n .G=H ILh�1iR / Š Lh�1in .R ÌH/IHHn .f�gIKtop

A;l1/ Š HG

n .G=H IKtopA;l1

/ Š Kn.A Ìl1 H/IHHn .f�gIKtop

A;r/ Š HGn .G=H IKtop

A;r/ Š Kn.A Ìr H/IHHn .f�gIKtop

A;m/ Š HGn .G=H IKtop

A;r/ Š Kn.A Ìm H/:

Here H and G act on R and A respectively via the given �-action, � W G ! �

and the inclusion H � G, Kn.R ÌH/ is the algebraic K-theory of the twistedgroup ring R ÌH , KHn.R ÌH/ is the homotopyK-theory of the twisted groupring R ÌH , Lh�1in .R ÌH/ is the algebraic L-theory with decoration h�1i ofthe twisted group ring with involution R Ì H , Kn.A Ìl1 H/ is the topologicalK-theory of the crossed product Banach algebra A Ìl1 H , Kn.A Ìr H/ isthe topological K-theory of the reduced crossed product C �-algebra A Ìr H ,and Kn.A Ìm H/ is the topological K-theory of the maximal crossed productC �-algebra A Ìm H ;


(iii) Let � W �0 ! �1 be a group homomorphism. Let R be a ring (with involution)andA be aC �-algebra on which �1 acts by structure preserving automorphisms.Let .G;�/ be a group over �0. Then in all cases the evaluation at .G;�/ of theequivariant homology theory over �0 associated to ��R or ��A respectivelyagrees with the evaluation at .G; � ı�/ of the equivariant homology theory over�1 associated to R or A respectively.

(iv) Suppose the group � acts on the rings (with involution) R and S or on theC �-algebras A and B respectively by structure preserving automorphisms. Let� W R ! S or � W A ! B be a �-equivariant homomorphism of rings (withinvolution) orC �-algebras respectively. Then � induces natural transformationsof homology theories over �

�‹� W H ‹�.�IKR/! H ‹�.�IKS /I�‹� W H ‹�.�IKHR/! H ‹�.�IKHS /I�‹� W H ‹�.�ILh�1iR /! H ‹�.�ILh�1iS /I�‹� W H ‹�.�IKtop

A;l1/! H ‹�.�IKtop

B;l1/I

�‹� W H ‹�.�IKtopA;r/! H ‹�.�IKtop

B;r/I�‹� W H ‹�.�IKtop

A;m/! H ‹�.�IKtopB;m/:

They are compatible with the identifications appearing in assertion (ii).

(v) Let � act on the C �-algebra A by structure preserving automorphisms. Wecan consider A also as a ring with structure preserving G-action. Then thereare natural transformations of equivariant homology theories with values in Z-modules over �

H ‹�.�IKA/! H ‹�.�IKHA/! H ‹�.�IKtopA;l1

/

! H ‹�.�IKtopA;m/! H ‹�.�IKtop

A;r/:

They are compatible with the identifications appearing in assertion (ii).

6.2 (Strong) continuity. Next we want to show

Lemma 6.2. Suppose that we are given a group � and a ring R (with involution) or aC �-algebra A respectively on which G acts by structure preserving automorphisms.

Then the homology theories with values in Z-modules over �

H ‹�.�IKR/;H‹�.�IKHR/; H ‹�.�ILh�1iR /; H ‹�.�IKtop

A;l1/; and H ‹�.�IKtop

A;m/

(see Theorem 6.1) are strongly continuous in the sense of Definition 3.3, whereas

H ‹�.�IKtopA;r/

is only continuous.


Proof. We begin with H ‹�.�IKR/ and H ‹�.�IKHR/. We have to show for everydirected systems of groups fGi j i 2 I g with G D colimi2I Gi together with a map� W G ! � that the canonical maps

colimi2I Kn.R ÌGi /! Kn.R ÌG/Icolimi2I KHn.R ÌGi /! KHn.R ÌG/;

are bijective for all n 2 Z. Obviously R Ì G is the colimit of rings colimi2I R Ì Gi .Now the claim follows for Kn.R ÌG/ for n � 0 from [36, (12) on page 20].

Using the Bass–Heller–Swan decomposition one gets the results forKn.RÌG/ forall n 2 Z and that the map

colimi2I N pKn.R ÌGi /! N pKn.R ÌG/

is bijective for all n 2 Z and all p 2 Z; p � 1 for the Nil-groups N pKn.RG/ definedby Bass [8, XII]. Now the claim for homotopy K-theory follows from the spectralsequence due to Weibel [41, Theorem 1.3].

Next we treatH ‹�.�ILh�1iR /. We have to show for every directed systems of groupsfGi j i 2 I g withG D colimi2I Gi together with a map � W G ! � that the canonicalmap

colimi2I Lh�1in .R ÌGi /! Lh�1in .R �G/is bijective for all n 2 Z. Recall from [37, Definition 17.1 and Definition 17.7] that

Lh�1in .R ÌG/ D colimm!1Lh�min .R ÌG/ILh�min .R ÌG/ D coker

�Lh�mC1inC1 .R ÌG/! L

h�mC1inC1 .R ÌGŒZ/

�for m � 0.

Since Lh1in .R ÌG/ is Lhn.R ÌG/, it suffices to show that

!n W colimi2I Lhn.R ÌGi /! Lhn.R ÌG/

is bijective for all n 2 Z. We give the proof of surjectivity for n D 0 only, the proofsof injectivity for n D 0 and of bijectivity for the other values of n are similar.

The ring RÌG is the colimit of rings colimi2I RÌGi . Let i W RÌGi ! RÌGand �i;j W R Ì Gi ! R Ì Gj for i; j 2 I; i � j be the structure maps. One candefine R ÌG as the quotient of

ì2I R ÌGi= , where x 2 R ÌGi and y 2 R ÌGj

satisfy x y if and only if �i;k.x/ D �j;k.y/ holds for some k 2 I with i; j � k.The addition and multiplication is given by adding and multiplying representativesbelonging to the same R ÌGi . LetM.m; nIR ÌG/ be the set of .m; n/-matrices withentries in R Ì G. Given Ai 2 M.m; nIR Ì Gi /, define �i;j .Ai / 2 M.m; nIR Ì Gj /and i .Ai / 2M.m; nIRÌG/ by applying �i;j and i to each entry of the matrix Ai .We need the following key properties which follow directly from inspecting the modelfor the colimit above:

(i) Given A 2M.m; nIR ÌG/, there exists i 2 I and Ai 2M.m; nIR ÌGi / with i .Ai / D A.


(ii) GivenAi 2M.m; nIRÌGi / andAj 2M.m; nIRÌGj /with i .Ai / D j .Aj /,there exists k 2 I with i; j � k and �i;k.Ai / D �j;k.Aj /.

An element ŒA in Lh0.R Ì G/ is represented by a quadratic form on a finitelygenerated free R Ì G-module, i.e., a matrix A 2 GLn.R Ì G/ for which there existsa matrix B 2 M.n; nIR Ì G/ with A D B C B�, where B� is given by transposingthe matrix B and applying the involution of R elementwise. Fix such a choice ofa matrix B . Choose i 2 I and Bi 2 M.n; nIR Ì Gi / with i .Bi / D B . Then i .Bi C B�i / D A is invertible. Hence we can find j 2 I with i � j such thatAj WD �i;j .Bi C B�j / is invertible. Put Bj D �i;j .Bi /. Then Aj D Bj C B�j and

j .Aj / D A. Hence Aj defines an element ŒAj 2 Lhn.R Ì Gj / which is mapped toŒA under the homomorphism Lhn.R Ì Gj /! Ln.R Ì G/ induced by j . Hence themap !0 of (6.2) is surjective.

Next we deal with H ‹�.�IKtopA;l1

/. We have to show for every directed systems ofgroups fGi j i 2 I g with G D colimi2I Gi together with a map � W G ! � that thecanonical map

colimi2I Kn.A Ìl1 Gi /! Kn.A Ìl1 G/is bijective for all n 2 Z. Since topologicalK-theory is a continuous functor, it sufficesto show that the colimit (or sometimes also called inductive limit) of the system ofBanach algebras fA Ìl1 Gi j i 2 I g in the category of Banach algebras with normdecreasing homomorphisms is A Ìl1 G. So we have to show that for any Banachalgebra B and any system of (norm deceasing) homomorphisms of Banach algebras˛i W AÌl1Gi ! B compatible with the structure mapsAÌl1�i;j W AÌl1Gi ! AÌl1Gjthere exists precisely one homomorphism of Banach algebras ˛ W A Ìl1 G ! B withthe property that its composition with the structure mapAÌl1 i W AÌl1Gi ! AÌl1Gis ˛i for i 2 I .

It is easy to see that in the category of C-algebras the colimit of the system fAÌGi ji 2 I g is AÌG with structure maps AÌ i W AÌGi ! AÌG. Hence the restrictionsof the homomorphisms ˛i to the subalgebrasAÌGi yields a homomorphism of centralC-algebras ˛0 W AÌG ! B uniquely determined by the property that the compositionof ˛0 with the structure mapAÌ i W AÌGi ! AÌG is ˛i jAÌGi for i 2 I . If ˛ exists,its restriction to the dense subalgebraAÌG has to be ˛0. Hence ˛ is unique if it exists.Of course we want to define ˛ to be the extension of ˛0 to the completion A Ìl1 G ofAÌG with respect to the l1-norm. So it remains to show that ˛0 W AÌG ! B is normdecreasing. Consider an element u 2 A Ì G which is given by a finite formal sumu DP

g2F ag �g, where F � G is some finite subset ofG and ag 2 A for g 2 F . Wecan choose an index j 2 I and a finite set F 0 � Gj such that j jF 0 W F 0 ! F is one-to-one. For g 2 F let g0 2 F 0 denote the inverse image of g under this map. Considerthe element v DP

g02F 0 ag � g0 in A Ì Gj . By construction we have A Ì j .v/ D uand kvk D kuk DPn

iD1 kaik. We conclude

k˛0.u/k D k˛0 ı .A Ì j /.v/k D k j .v/k � kvk D kuk:The proof for H ‹�.�IKtop

A;m/ follows similarly, using the fact that by definition of thenorm onAÌmG every-homomorphism ofAÌG into aC �-algebraB extends uniquely


to A Ìm G. The proof for the continuity of H ‹�.�IKtopA;r/ follows from Theorem 4.1

in [10].

Notice that we have proved all promised results of the introduction as soon as wehave completed the proof of Theorem 6.1 which we have used as a black box so far.

7 From spectra over groupoids to equivariant homology theories

In this section we explain how one can construct equivariant homology theories fromspectra over groupoids.

A spectrum E D f.E.n/; .n// j n 2 Zg is a sequence of pointed spaces fE.n/ jn 2 Zg together with pointed maps called structure maps.n/ W E.n/^S1 ! E.nC1/.A (strong) map of spectra (sometimes also called function in the literature) f W E! E0is a sequence of maps f .n/ W E.n/ ! E 0.n/ which are compatible with the structuremaps .n/, i.e., we have f .nC 1/ ı .n/ D 0.n/ ı .f .n/^ idS1/ for all n 2 Z. Thisshould not be confused with the notion of a map of spectra in the stable category (see[1, III.2.]). Recall that the homotopy groups of a spectrum are defined by

�i .E/ WD colimk!1 �iCk.E.k//;

where the system �iCk.E.k// is given by the composition

�iCk.E.k//S�! �iCkC1.E.k/ ^ S1/ �.k/��! �iCkC1.E.k C 1//

of the suspension homomorphism and the homomorphism induced by the structuremap. We denote by Spectra the category of spectra.

A weak equivalence of spectra is a map f W E ! F of spectra inducing an isomor-phism on all homotopy groups.

Given a small groupoid G , denote by Groupoids # G the category of small group-oids over G , i.e., an object is a functor F0 W G0 ! G with a small groupoid as sourceand a morphism fromF0 W G0 ! G toF1 W G1 ! G is a functorF W G0 ! G1 satisfyingF1 ıF D F0. We will consider a group � as a groupoid with one object and � as set ofmorphisms. An equivalence F W G0 ! G1 of groupoids is a functor of groupoids F forwhich there exists a functor of groupoids F 0 W G1 ! G0 such that F 0 ıF and F ıF 0 arenaturally equivalent to the identity functor. A functorF W G0 ! G1 of small groupoids isan equivalence of groupoids if and only if it induces a bijection between the isomorphismclasses of objects and for any object x 2 G0 the map autG0.x/! autG1.F.x// inducedby F is an isomorphism of groups.

Lemma 7.1. Let � be a group. Consider a covariant functor

E W Groupoids # � ! Spectra

which sends equivalences of groupoids to weak equivalences of spectra.


Then we can associate to it an equivariant homology theory H ‹�.;�IE/ (with valuesin Z-modules) over � such that for every group .G;�/ over � and subgroup H � Gwe have a natural identification

HHn .f�gIE/ D HG

n .G=H;E/ D �n.E.H//:If T W E ! F is a natural transformation of such functors Groupoids # � !

Spectra, then it induces a transformation of equivariant homology theories over �

H ‹�.�IT/ W H ‹�.;�IE/! H ‹�.;�IF/such that for every group .G;�/ over � and subgroup H � G the homomorphismHHn .f�gIT/ W HH

n .f�gIE/ ! HHn .f�gIF/ agrees under the identification above with

�n.T.H// W �n.E.H//! �n.F.H//.

Proof. We begin with explaining how we can associate to a group .G;�/ over � aG-homology theory HG� .�IE/ with the property that for every subgroup H � G wehave an identification

HGn .G=H;E/ D �n.E.H//:

We just follow the construction in [13, Section 4]. Let Or.G/ be the orbit category ofG,i.e., objects are homogenous spacesG=H and morphisms areG-maps. Given aG-setS ,the associated transport groupoid tG.S/ hasS as set of objects and the set of morphismsfrom s0 2 S to s1 2 S consists of the subset fg 2 G j gs1 D s2g of G. Compositionis given by the group multiplication. A G-map of sets induces a functor betweenthe associated transport groupoids in the obvious way. In particular the projectionG=H ! G=G induces a functor of groupoids prS W tG.S/ ! tG.G=G/ D G. ThustG.S/ becomes an object in Groupoids # � by the composite � ı prS . We obtaina covariant functor tG W Or.G/ ! Groupoids # � . Its composition with the givenfunctor E yields a covariant functor

EG WD E ı tG W Or.G/! Spectra:

Now defineHG� .X;AIE/ WD HG� .�IEG/;

where HG� .�IEG/ is the G-homology theory which is associated to EG W Or.G/ !Spectra and defined in [13, Section 4 and 7]. Namely, ifX is aG-CW-complex, we canassign to it a contravariant functor mapG.G=‹;X/ W Or.G/ ! Spaces sending G=Hto mapG.G=H;X/ D XH and put HG

n .X IEG/ WD �n.mapG.G=‹;X/C Ôr.G/ EG/for the spectrum mapG.G=‹;X/C Ôr.G/ EG (which is denoted in [13] bymapG.G=‹;X/C ˝Or.G/ EG).

Next we have to explain the induction structure. Consider a group homomorphism˛ W .H; �/! .G;�/ of groups over � and anH -CW-complexX . We have to constructa homomorphism

HHn .X IE/! HG

n .˛�X IE/:


This will be done by constructing a map of spectra

mapH .H=‹;X/C Ôr.H/ EH ! mapG.G=‹; ˛�X/C Ôr.G/ EG :

We follow the constructions in [13, Section 1]. The homomorphism ˛ induces a co-variant functor Or.˛/ W Or.H/ ! Or.G/ by sending H=L to ˛�.H=L/ D G=˛.L/.Given a contravariant functor Y W Or.H/ ! Spaces, we can assign to it its inductionwith Or.˛/ which is a contravariant functor ˛�Y W Or.G/ ! Spaces. Given a con-travariant functor Z W Or.G/ ! Spaces, we can assign to it its restriction which isthe contravariant functor ˛�Z WD Z ı Or.˛/ W Or.G/ ! Spaces. Induction ˛� and˛� form an adjoint pair. Given an H -CW-complex X , there is a natural identifica-tion ˛�.mapH .H=‹;X// D mapG.G=‹; ˛�X/. Using [13, Lemma 1.9] we get for anH -CW-complex X a natural map of spectra

mapH .H=‹;X/C Ôr.H/ ˛�EG ! mapG.G=‹; ˛�X/C Ôr.G/ EG :

Given anH -set S , we obtain a functor of groupoids tH .S/! tG.˛�S/ sending s 2 Sto .1; s/ 2 G �˛ S and a morphism in tH .S/ given by a group element h to the onein tG.˛�S/ given by ˛.h/. This yields a natural transformation of covariant functorsOr.H/ ! Groupoids # � from tH ! tG ı Or.˛/. Composing with the functor Egives a natural transformation of covariant functors Or.H/ ! Spectra from EH to˛�EG . It induces a map of spectra

mapH .H=‹;X/C Ôr.H/ EH ! mapH .H=‹;X/C Ôr.H/ ˛�EG :

Its composition with the maps of spectra constructed beforehand yields the desired mapof spectra mapH .H=‹;X/C ˝Or.H/ EH ! mapG.G=‹; ˛�X/C ˝Or.G/ EG .

We omit the straightforward proof that the axioms of an induction structure aresatisfied. This finishes the proof of Theorem 6.1.

The statement about the natural transformation T W E! F is obvious.

8 Some K -theory spectra associated to groupoids

The last step in completing the proof of Theorem 6.1 is to prove the following The-orem 8.1 because then we can apply it in combination with Lemma 7.1. (Actuallywe only need the version of Theorem 8.1, where G is given by a group � .) LetGroupoidsfinker # G be the subcategory of Groupoids # G which has the same objectsand for which a morphism from F0 W G0 ! G to F1 W G1 ! G given by a functorF W G0 ! G1 satisfying F1 ı F D F0 has the property that for every object x 2 G0the group homomorphism autG0.x/ ! autG1.F.x// induced by F has a finite ker-nel. Denote by Rings, -Rings, and C �-C �Algebras the categories of rings, rings withinvolution and C �-algebras.

Theorem 8.1. Let G be a fixed groupoid. Let R W G ! Rings, R W G ! -Rings, orA W G ! C �Algebras respectively be a covariant functor. Then there exist covariant


functors

KR W Groupoids # G ! SpectraIKHR W Groupoids # G ! SpectraI

Lh�1iR W Groupoids # G ! SpectraIKtopA;l1W Groupoids # G ! SpectraI

KtopA;r W Groupoidsfinker # G ! SpectraI

KtopA;m W Groupoids # G ! Spectra;

together with natural transformations

I1 W K! KHII2 W KH! Ktop

A;l1I

I3 W KtopA;l1! Ktop

A;mII4 W Ktop

A;m ! KtopA;r ;

of functors from Groupoids # G or Groupoidsfinker # G respectively to Spectra suchthat the following holds:

(i) Let Fi W Gi ! G be objects for i D 0; 1 and F W F0 ! F1 be a morphismbetween them in Groupoids # G or Groupoidsfinker # G respectively such thatthe underlying functor of groupoids F W G0 ! G1 is an equivalence of groupoids.Then the functors send F to a weak equivalences of spectra.

(ii) Let F0 W G0 ! G be an object in Groupoids # G or Groupoidsfinker # G re-spectively such that the underlying groupoid G0 has only one object x. LetG D morG0.x; x/ be its automorphisms group. We obtain a ring R.y/, a ringR.y/ with involution, or a C �-algebra B.y/ with G-operation by structure pre-serving maps from the evaluation of the functorR orA respectively at y D F.x/.Then:

�n.KR.F // D Kn.R.y/ ÌG/I�n.KHR.F // D KHn.R.y/ ÌG/I�n.L

h�1iR .F // D Lh�1in .R.y/ ÌG/I

�n.KtopA.y/;l1

.F // D Kn.A.y/ Ìl1 G/I�n.K

topA.y/;r

.F // D Kn.A.y/ Ìr G/I�n.K

topA.y/;m

.F // D Kn.A.y/ Ìm G/;

whereKn.R.y/ÌG/ is the algebraicK-theory of the twisted group ringR.y/ÌG,KHn.R.y/ ÌG/ is the homotopy K-theory of the twisted group ring R.y/ ÌG,


Lh�1in .R.y/ÌG/ is the algebraicL-theory with decoration h�1i of the twisted

group ring with involution R.y/ Ì G, Kn.A.y/ Ìl1 G/ is the topological K-theory of the crossed product Banach algebra A.y/ Ìl1 G, Kn.A.y/ Ìr G/ isthe topologicalK-theory of the reduced crossed product C �-algebraA.y/Ìr G,andKn.A.y/ÌmG/ is the topologicalK-theory of the maximal crossed productC �-algebra A.y/ Ìm G.

The natural transformations I1, I2, I3 and I4 become under this identificationsthe obvious change of rings and theory homomorphisms.

(iii) These constructions are in the obvious sense natural inR andA respectively andin G .

We defer the details of the proof of Theorem 8.1 in [2]. Its proof requires some workbut there are many special cases which have already been taken care of. If we wouldnot insist on groupoids but only on groups as input, these are the standard algebraicK-and L-theory spectra or topological K-theory spectra associated to group rings, groupBanach algebras and group C �-algebras. The construction for the algebraic K- andL-theory and the topological K-theory in the case, where G acts trivially on a ring Ror a C �-algebra are already carried out or can easily be derived from [4], [13], and [24]except for the case of a Banach algebra. The case of theK-theory spectrum associatedto an additive category with G-action has already been carried out in [7]. The mainwork which remains to do is to treat the Banach case and to construct the relevantnatural transformation from KH to Ktop

A;l1

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[8] H. Bass, Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968.

[9] P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and K-theory ofgroup C�-algebras, in C�-algebras: 1943–1993 (San Antonio, TX, 1993), Amer. Math.Soc., Providence, RI, 1994, 240–291.


[10] P. Baum, S. Millington, and R. Plymen, Local-global principle for the Baum-Connes con-jecture with coefficients, K-Theory 28 (2003), 1–18.

[11] B. H. Bowditch, Notes on Gromov’s hyperbolicity criterion for path-metric spaces, inGroup theory from a geometrical viewpoint (Trieste, 1990), World Scientific Publishing,River Edge, NJ, 1991, 64–167.

[12] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, GrundlehrenMath. Wiss. 319, Springer-Verlag, Berlin 1999.

[13] J. F. Davis and W. Lück, Spaces over a category and assembly maps in isomorphismconjectures in K- and L-theory, K-Theory 15 (1998), 201–252.

[14] P. de la Harpe and A. Valette, La propriété .T / de Kazhdan pour les groupes localementcompacts (avec un appendice de Marc Burger), Astérisque 175 (1989).

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[16] F. T. Farrell and P. A. Linnell, K-theory of solvable groups, Proc. London Math. Soc. (3)87 (2003), 309–336.

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Coarse and equivariant co-assembly maps

Heath Emerson and Ralf Meyer�

1 Introduction

This is a sequel to the articles [5], [7], which deal with a coarse co-assembly map thatis dual to the usual coarse assembly map. Here we study an equivariant co-assemblymap that is dual to the Baum–Connes assembly map for a group G.

A rather obvious choice for such a dual map is the map

p�EG W KKG� .C;C/! RKKG� .EGIC;C/ (1)

induced by the projection pEG W EG ! point. This map and its application to theNovikov conjecture go back to Kasparov ([10]). Nevertheless, (1) is not quite themap that we consider here. Our map is closely related to the coarse co-assembly mapof [5]. It is an isomorphism if and only if the Dirac-dual-Dirac method applies to G.Hence there are many cases – groups with � ¤ 1 – where our co-assembly map is anisomorphism and (1) is not.

Most of our results only work if the groupG is (almost) totally disconnected and hasa G-compact universal proper G-space EG. We impose this assumption throughoutthe introduction.

First we briefly recall some of the main ideas of [5], [7]. The new ingredient inthe coarse co-assembly map is the reduced stable Higson corona cred.X/ of a coarsespace X . Its definition resembles that of the usual Higson corona, but its K-theorybehaves much better. The coarse co-assembly map is a map

� W K�C1�cred.X/

�! KX�.X/; (2)

where KX�.X/ is a coarse invariant of X that agrees with K�.X/ if X is uniformlycontractible.

If jGj is the coarse space underlying a groupG, then there is a commuting diagram

KKG��C; C0.G/

� p�

EG ��

��

Š��

RKKG��EGIC; C0.G/

��

Š��

K�C1�cred.jGj/� �

�� KX�.jGj/.

(3)

�This research was partially carried out at the Westfälische Wilhelms-Universität Münster and supportedby the EU-Network Quantum Spaces and Noncommutative Geometry (Contract HPRN-CT-2002-00280) andthe Deutsche Forschungsgemeinschaft (SFB 478).

72 H. Emerson and R. Meyer

In this situation, KX�.jGj/ Š K�.EG/ because jGj is coarsely equivalent to EG, whichis uniformly contractible. The commuting diagram (3), coupled with the reformulationof the Baum–Connes assembly map in [11], is the source of the relationship betweenthe coarse co-assembly map and the Dirac-dual-Dirac method mentioned above.

If G is a torsion-free discrete group with finite classifying space BG, the coarseco-assembly map is an isomorphism if and only if the Dirac-dual-Dirac method appliesto G. A similar result for groups with torsion is available, but this requires workingequivariantly with respect to compact subgroups of G.

In this article, we work equivariantly with respect to the whole groupG. The actionof G on its underlying coarse space jGj by isometries induces an action on cred.G/.We consider a G-equivariant analogue

� W Ktop�C1

�G; cred.jGj/�! K�.C0.EG/ ÌG/ (4)

of the coarse co-assembly map (2); here Ktop� .G;A/ denotes the domain of the Baum–Connes assembly map for G with coefficients A. We avoid K�.cred.X/ Ì G/ andK�.cred.X/ÌrG/ because we can say nothing about these two groups. In contrast, thegroup Ktop�

�G; cred.jGj� is much more manageable. The only analytical difficulties in

this group come from coarse geometry.There is a commuting diagram similar to (3) that relates (4) to equivariant Kasparov

theory. To formulate this, we need some results of [11]. There is a certain G-C�-alge-bra P and a class D 2 KKG.P;C/ called Dirac morphism such that the Baum–Connesassembly map for G is equivalent to the map

K��.A˝ P/ Ìr G/! K�.A Ìr G/

induced by Kasparov product with D. The Baum–Connes conjecture holds for G withcoefficients in P˝A for any A. The Dirac morphism is a weak equivalence, that is, itsimage in KKH .P;C/ is invertible for each compact subgroup H of G.

The existence of the Dirac morphism allows us to localise the (triangulated) categoryKKG at the multiplicative system of weak equivalences. The functor from KKG to itslocalisation turns out to be equivalent to the map

p�EG W KKG.A;B/! RKKG.EGIA;B/:One of the main results of this paper is a commuting diagram

Ktop�C1

�G; cred.jGj/� ��

��

Š��

K�.EG/��

Š��

KKG� .C;P/p�

EG �� RKKG� .EGIC;P/.(5)

In other words, the equivariant coarse co-assembly (4) is equivalent to the map

p�EG W KKG� .C;P/! RKKG� .EGIC;P/:

Coarse and equivariant co-assembly maps 73

This map is our proposal for a dual to the Baum–Connes assembly map.We should justify why we prefer the map (4) over (1). Both maps have isomorphic

targets:

RKKG� .EGIC;P/ Š RKKG� .EGIC;C/ Š K�.C0.EG/ ÌG/:

Even in the usual Baum–Connes assembly map, the analytical side involves a choicebetween full and reduced group C�-algebras and crossed products. Even though thefull group C�-algebra has better functoriality properties and is sometimes preferredbecause it gives potentially finer invariants, the reduced one is used because its K-the-ory is closer to Ktop� .G/. In formulating a dual version of the assembly map, weare faced with a similar situation. Namely, the topological object that is dual toKtop� .G/ is RKKG� .EGIC;C/. For the analytical side, we have some choices; weprefer KKG� .C;P/ over KKG� .C;C/ because the resulting co-assembly map is an iso-morphism in more cases.

Of course, we must check that this choice is analytical enough to be useful forapplications. The most important of these is the Novikov conjecture. Elements ofRKKG� .EGIC;C/ yield maps Ktop� .G/! Z, which are analogous to higher signatures.In particular, (4) gives rise to such objects. The maps Ktop� .G/! Z that come from aclass in the range of (1) are known to yield homotopy invariants for manifolds becausethere is a pairing between KKG� .C;C/ and K�.C �maxG/ (see [8]). But since (4) factorsthrough (1), the former also produces homotopy invariants.

In particular, surjectivity of (4) implies the Novikov conjecture for G. More istrue: since KKG.C;P/ is the home of a dual-Dirac morphism, (5) yields that G has adual-Dirac morphism and hence a � -element if and only if (4) is an isomorphism. Thisobservation can be used to give an alternative proof of the main result of [7].

We call elements in the range of (4) boundary classes. These automatically form agraded ideal in the Z=2-graded unital ring RKKG� .EGIC;C/. In contrast, the rangeof the unital ring homomorphism (1) need not be an ideal because it always containsthe unit element of RKKG� .EGIC;C/. We describe two important constructions ofboundary classes, which are related to compactifications and to the proper Lipschitzcohomology of G studied in [3], [4].

Let EG � Z be a G-equivariant compactification of EG that is compatible withthe coarse structure in a suitable sense. Since there is a map

Ktop��G;C.Z n EG/

�! Ktop��G; cred.EG/

�;

we get boundary classes from the boundary Z n EG. This construction also showsthat G has a dual-Dirac morphism if it satisfies the Carlsson–Pedersen condition. Thisimproves upon a result of Nigel Higson ([9]), which shows split injectivity of theBaum–Connes assembly map with coefficients under the same assumptions.

Although we have discussed only KKG� .C;P/ so far, our main technical result ismore general and can also be used to construct elements in Kasparov groups of theform KKG�

�C; C0.X/

�for suitableG-spacesX . IfX is a properG-space, then we can

use such classes to construct boundary classes in RKKG� .EGIC;C/. This provides a


K-theoretic counterpart of the proper Lipschitz cohomology of G defined by Connes,Gromov, and Moscovici in [3]. Our approach clarifies the geometric parts of severalconstructions in [3]; thus we substantially simplify the proof of the homotopy invarianceof Gelfand–Fuchs cohomology classes in [3].

2 Preliminaries

2.1 Dirac-dual-Dirac method and Baum–Connes conjecture. First, we recall theDirac-dual-Dirac method of Kasparov and its reformulation in [11]. This is a techniquefor proving injectivity of the Baum–Connes assembly map

� W Ktop� .G;B/! K��C �r .G;B/

�; (6)

where G is a locally compact group and B is a C�-algebra with a strongly continuousaction of G or, briefly, a G-C�-algebra.

This method requires a proper G-C�-algebra A and classes

d 2 KKG.A;C/; � 2 KKG.C; A/; � WD �˝A d 2 KKG.C;C/;

such that p�EG.�/ D 1C in RKKG.EGIC;C/. If these data exist, then the Baum–

Connes assembly map (6) is injective for allB . If, in addition, � D 1C in KKG.C;C/,then the Baum–Connes assembly map is invertible for all B , so that G satisfies theBaum–Connes conjecture with arbitrary coefficients.

LetA andB beG-C�-algebras. An elementf 2 KKG.A;B/ is called a weak equiv-alence in [11] if its image in KKH .A;B/ is invertible for each compact subgroup Hof G.

The following theorem contains some of the main results of [11].

Theorem 1. Let G be a locally compact group. Then there is a G-C�-algebra P anda class D 2 KKG.P;C/ called Dirac morphism such that

(a) D is a weak equivalence;

(b) the Baum–Connes conjecture holds with coefficients in A˝ P for any A;

(c) the assembly map (6) is equivalent to the map

D� W K�.A˝ P Ìr G/! K�.A Ìr G/I

(d) the Dirac-dual-Dirac method applies to G if and only if there is a class � 2KKG.C;P/ with �˝C D D 1A, if and only if the map

D� W KKG� .C;P/! KKG� .P;P/; x 7! D ˝ x; (7)

is an isomorphism.


Whereas [7] studies the invertibility of (7) by relating it to (2), here we are going tostudy the map (7) itself.

It is shown in [11] that the localisation of the category KKG at the weak equivalencesis isomorphic to the category RKKG.EG/ whose morphism spaces are the groupsRKKG.EGIA;B/ as defined by Kasparov in [10]. This statement is equivalent to theexistence of a Poincaré duality isomorphism

KKG� .A˝ P; B/ Š RKKG� .EGIA;B/ (8)

for allG-C�-algebrasA andB (this notion of duality is analysed in [6]). The canonicalfunctor from KKG to the localisation becomes the obvious functor

p�EG W KKG.A;B/! RKKG.EGIA;B/:Since D is a weak equivalence, p�

EG.D/ is invertible. Hence the maps in the fol-

lowing commuting square are isomorphisms for all G-C�-algebras A and B:

RKKG� .EGIA;B ˝ P/ ŠD� ��

Š D�

��

RKKG� .EGIA;B/Š D�

��

RKKG� .EGIA˝ P; B ˝ P/ ŠD� �� RKKG� .EGIA˝ P; B/.

Together with (8) this implies

KKG� .A˝ P; B/ Š KKG� .A˝ P; B ˝ P/:

In the following, it will be useful to turn the isomorphism

Ktop� .G;A/ Š K��.A˝ P/ Ìr G

�in Theorem 1.(c) into a definition.

2.2 Group actions on coarse spaces. LetG be a locally compact group and letX bea right G-space and a coarse space. We always assume that G acts continuously andcoarsely on X , that is, the set f.xg; yg/ j g 2 K; .x; y/ 2 Eg is an entourage for anycompact subset K of G and any entourage E of X .

Definition 2. We say that G acts by translations on X if f.x; gx/ j x 2 X; g 2 Kgis an entourage for all compact subsets K � G. We say that G acts by isometries ifevery entourage of X is contained in a G-invariant entourage.

Example 3. Let G be a locally compact group. Then G has a unique coarse structurefor which the right translation action is isometric; the corresponding coarse space isdenoted jGj. The generating entourages are of the form[

g2GKg �Kg D f.xg; yg/ j g 2 G; x; y 2 Kg

for compact subsets K of G. The left translation action is an action by translations forthis coarse structure.


Example 4. More generally, any proper,G-compactG-spaceX carries a unique coarsestructure for which G acts isometrically; its entourages are defined as in Example 3.With this coarse structure, the orbit map G ! X , g 7! g � x, is a coarse equivalencefor any choice of x 2 X . If the G-compactness assumption is omitted, the result is a� -coarse space. We always equip a proper G-space with this additional structure.

2.3 The stable Higson corona. We next recall the definition of the stable Higsoncorona of a coarse space X from [5], [7]. Let D be a C�-algebra.

Let M.D ˝ K/ be the multiplier algebra of D ˝ K, and let xBred.X;D/ be theC�-algebra of norm-continuous, bounded functions f W X ! M.D ˝ K/ for whichf .x/ � f .y/ 2 D ˝K for all x; y 2 X . We also let

Bred.X;D/ WD xBred.X;D/=C0.X;D ˝K/:

Definition 5. A function f 2 xBred.X;D/ has vanishing variation if the functionE 3 .x; y/ 7! kf .x/ � f .y/k vanishes at1 for any closed entourage E � X �X .

The reduced stable Higson compactification of X with coefficients D is the subal-gebra Ncred.X;D/ � xBred.X;D/ of vanishing variation functions. The quotient

cred.X;D/ WD Ncred.X;D/=C0.X;D ˝K/ � Bred.X;D/

is called reduced stable Higson corona of X . This defines a functor on the coarsecategory of coarse spaces: a coarse map f W X ! X 0 induces a map cred.X 0;D/ !cred.X;D/, and two maps X ! X 0 induce the same map cred.X 0;D/! cred.X;D/

if they are close. Hence a coarse equivalence X ! X 0 induces an isomorphismcred.X 0;D/ Š cred.X;D/.

For some technical purposes, we must allow unions X D SXn of coarse spaces

such that the embeddings Xn ! XnC1 are coarse equivalences; such spaces are called� -coarse spaces. The main example is the Rips complex P .X/ of a coarse space X ,which is used to define its coarse K-theory. More generally, if X is a proper but notG-compact G-space, then X may be endowed with the structure of a � -coarse space.For coarse spaces of the form jGj for a locally compact group G with a G-compactuniversal properG-space EG, we may use EG instead of P .X/ because EG is coarselyequivalent toG and uniformly contractible. Therefore, we do not need � -coarse spacesmuch; they only occur in Lemma 7.

It is straightforward to extend the definitions of Ncred.X;D/ and cred.X;D/ to� -coarse spaces (see [5], [7]). Since we do not use this generalisation much, we omitdetails on this.

LetH be a locally compact group that acts coarsely and properly onX . It is crucialfor us to allow non-compact groups here, whereas [7] mainly needs equivariance forcompact groups. LetD be anH -C�-algebra, and let KH WD K.`2N˝L2H/. ThenHacts on xBred.X;D ˝KH / by

.h � f /.x/ WD h � �f .xh/�;


where we use the obvious action of H on D ˝ KH and its multiplier algebra. Theaction of H on xBred.X;D ˝KH / need not be continuous; we let xBred

H .X;D/ be thesubalgebra of H -continuous elements in xBred.X;D ˝ KH /. We let Ncred

H .X;D/ bethe subalgebra of vanishing variation functions in xBred

H .X;D/. Both algebras containC0.X;D˝KH / as an ideal. The corresponding quotients are denoted by Bred

H .X;D/

and credH .X;D/. By construction, we have a natural morphism of extensions ofH -C�-

algebras

C0.X;D ˝KH / �� NcredH .X;D/ ��

��

credH .X;D/

��

C0.X;D ˝KH / �� xBredH .X;D/ �� Bred

H .X;D/.

(9)

Concerning the extension of this construction to � -coarse spaces, we only mentionone technical subtlety. We must extend the functor Ktop� .H; / from C�-algebras to� -H -C�-algebras. Here we use the definition

Ktop� .H;A/ Š K��.A˝ P/ Ìr H

�; (10)

where D 2 KKH .P;C/ is a Dirac morphism for H . The more traditional definitionas a colimit of KKG� .C0.X/; A/, where X � EG is G-compact, yields a wrong resultif A is a � -H -C�-algebra because colimits and limits do not commute.

Let H be a locally compact group, let X be a coarse space with an isometric,continuous, proper action of H , and let D be an H -C�-algebra. The H -equivariantcoarse K-theory KX�H .X;D/ of X with coefficients in D is defined in [7] by

KX�H .X;D/ WD Ktop��H;C0.P .X/;D/

�: (11)

As observed in [7], we have Ktop��H;C0.P .X/;D/

� Š K�.C0.P .X/;D/ Ì H/ be-cause H acts properly on P .X/.

For most of our applications, X will be equivariantly uniformly contractible forall compact subgroups K � H , that is, the natural embedding X ! P .X/ is aK-equivariant coarse homotopy equivalence. In such cases, we simply have

KX�H .X;D/ Š Ktop��H;C0.X;D/

�: (12)

In particular, this applies ifX is anH -compact universal properH -space (again, recallthat the coarse structure is determined by requiring H to act isometrically).

TheH -equivariant coarse co-assembly map forX with coefficients inD is a certainmap

�� W Ktop�C1

�H; cred

H .X;D/�! KX�H .X;D/

defined in [7]. In the special case where we have (12), this is simply the boundarymap for the extension C0.X;D ˝ KH / � Ncred

H .X;D/ � credH .X;D/. We are

implicitly using the fact that the functor Ktop� .H; /has long exact sequences for arbitraryextensions of H -C�-algebras, which is proved in [7] using the isomorphism

Ktop� .H;B/ WD K��.B ˝ P/ Ìr H

� Š K��.B ˝max P/ ÌH

�


and exactness properties of maximal C�-tensor products and full crossed products.There is also an alternative picture of the co-assembly map as a forget-control map,

providedX is uniformly contractible (see [7, §2.8]). We have the following equivariantversion of this result:

Proposition 6. Let G be a totally disconnected group with a G-compact universalproper G-space EG. Then the G-equivariant coarse co-assembly map for G is equiv-alent to the map

j� W Ktop�C1

�G; cred

G .EG;D/�! Ktop

�C1�G;Bred

G .EG;D/�

induced by the inclusion j W credG .EG;D/! Bred

G .EG;D/.

The equivalence of the two maps means that there is a natural commuting diagram

Ktop�C1

�G; cred

G .jGj;D/� ��

��

��

Š��

KXG� .jGj;D/��

Š��

Ktop�C1

�G; cred

G .EG;D/� j�

�� Ktop�C1

�G;Bred

G .EG;D/�.

Recall that j is induced by the inclusion NcredG .EG;D/! xBred

G .EG;D/, which exactlyforgets the vanishing variation condition. Hence j� is a forget-control map.

Proof. We may replace jGjby EG because EG is coarsely equivalent to jGj. The coarseK-theory of EG agrees with the usual K-theory of EG (see [7]). A slight elaborationof the proof of [7, Lemma 15] shows that

KH��xBred

G .EG;D/� Š KKH�

�C; xBred

G .EG;D/�

vanishes for all compact subgroups H of G. This yields Ktop��G; xBred

G .EG;D/� D 0

by a result of [2]. Now the assertion follows from the Five Lemma and the naturalityof the K-theory long exact sequence for (9) as in [7].

3 Classes in Kasparov theory from the stable Higson corona

In this section, we show how to construct classes in equivariant KK-theory from theK-theory of the stable Higson corona. The following lemma is our main technicaldevice:

Lemma 7. Let G and H be locally compact groups and let X be a coarse spaceequipped with commuting actions of G and H . Suppose that G acts by translationsand thatH acts properly and by isometries. LetA andD beH -C�-algebras, equippedwith the trivial G-action. We abbreviate

BX WD C0.X;D ˝KH ˝max A/ ÌH; EX WD .NcredH .X;D/˝max A/ ÌH


and similarly for P .X/ instead of X . There are extensions BX � EX � EX=BXand BP .X/ � EP .X/ � EP .X/=BP .X/ with

EP .X/=BP .X/ Š EX=BX Š .credH .X;D/˝max A/ ÌH;

and a natural commuting diagram

K�C1.EX=BX / @ ��

��

K�.BP .X//

�

��

KKG� .C; BX /p�

EG �� RKKG� .EGIC; BX /.(13)

Proof. The quotients EX=BX and EP .X/=BP .X/ are as asserted and agree becauseX ! P .X/ is a coarse equivalence and because maximal tensor products and fullcrossed products are exact functors in complete generality, unlike spatial tensor productsand reduced crossed products. We let @ be the K-theory boundary map for the extensionBP .X/ � EP .X/ � EX=BX .

Since we have a natural map credH .X;D/˝max A ! cred

H .X;D ˝max A/, we mayreplace the pair .D;A/ by .D ˝max A;C/ and omit A if convenient. Stabilising Dby KH , we can further eliminate the stabilisations.

First we lift the K-theory boundary map for the extension BX � EX � EX=BXto a map W K�C1.EX=BX / ! KKG� .C; BX /. The G-equivariance of the resultingKasparov cycles follows from the assumption that G acts on X by translations.

We have to distinguish between the cases � D 0 and � D 1. We only write downthe construction for � D 0. Since the algebra EX=BX is matrix-stable, K1.EX=BX /is the homotopy group of unitaries in EX=BX without further stabilisation. A cyclefor KKG0 .C; BX / is given by two G-equivariant Hilbert modules E˙ over BX and aG-continuous adjointable operator F W EC ! E� for which 1 � FF �, 1 � F �F andgF � F for g 2 G are compact; we take E˙ D BX and let F 2 EX � M.BX / bea lifting for a unitary u 2 EX=BX . Since G acts on X by translations, the inducedaction on Ncred.X;D/ and hence on EX=BX is trivial. Hence u is a G-invariant unitaryin EX=BX . For the lifting F , this means that

1 � FF �; 1 � F �F; gF � F 2 BX :Hence F defines a cycle for KKG0 .C; BX /. We get a well-defined map Œu� 7! ŒF � fromK1.EX=BX / to KKG0 .C; BX / because homotopic unitaries yield operator homotopicKasparov cycles.

Next we have to factor the map p�EGı in (13) through K0.BP .X//. The main

ingredient is a certain continuous map Nc W EG�X ! P .X/. We use the same descrip-tion of P .X/ as in [7] as the space of positive measures on X with 1=2 < �.X/ � 1;this is a � -coarse space in a natural way, we write it as P .X/ DS

Pd .X/.There is a function c W EG ! RC for which

REGc.�g/ dg D 1 for all � 2 EG

and supp c \ Y is compact for G-compact Y � EG. If � 2 EG, x 2 X , then the


condition

h Nc.�; x/; ˛i WDZG

c.�g/˛.g�1x/ dg

for ˛ 2 C0.X/ defines a probability measure onX . Since such measures are containedin P .X/, Nc defines a map Nc W EG �X ! P .X/. This map is continuous and satisfiesNc.�g; g�1xh/ D Nc.�; x/h for all g 2 G, � 2 EG, x 2 X , h 2 H .

For a C�-algebra Z, let C.EG;Z/ be the � -C�-algebra of all continuous func-tions f W EG ! Z without any growth restriction. Thus C.EG;Z/ D lim �C.K;Z/,where K runs through the directed set of compact subsets of EG.

We claim that . Nc�f /.�/.x/ WD f� Nc.�; x/� for f 2 C0.P .X/;D/ defines a con-

tinuous �-homomorphism

Nc� W C0.P .X/;D/! C�EG;C0.X;D/

�:

If K � EG is compact, then there is a compact subset L � G such that c.� � g/ D 0for � 2 K and g … L. Hence Nc.�; x/ is supported in L�1x for � 2 K. Since G actson X by translations, such measures are contained in a filtration level Pd .X/. HenceNc�.f / restricts to a C0-function K � X ! D for all f 2 C0.P .X/;D/. This provesthe claim. Since Nc is H -equivariant and G-invariant, we get an induced map

BP .X/ D C0.P .X/;D/ ÌH ! .C�EG;C0.X;D/

�ÌH/G D C.EG;BX /G ;

whereZG � Z denotes the subalgebra ofG-invariant elements. We obtain an induced�-homomorphism between the stable multiplier algebras as well.

An element of K0�BP .X// is represented by a self-adjoint bounded multiplier F 2

M.BP .X/ ˝K/ such that 1 � FF � and 1 � F �F belong to BP .X/ ˝K. Now QF WDNc�.F / is aG-invariant bounded multiplier ofC.EG;BX˝K/ and hence aG-invariantmultiplier of C0.EG;BX ˝ K/, such that ˛ � .1 � QF QF �/ and ˛ � .1 � QF � QF / belongto C0.EG;BX ˝ K/ for all ˛ 2 C0.EG/. This says exactly that QF is a cycle forRKKG0 .EGIC; BX /. This construction provides the natural map

� W K0.BP .X//! RKKG0 .EGIC; BX /:Finally, a routine computation, which we omit, shows that the two images of a uni-

tary u 2 EX=BX differ by a compact perturbation. Hence the diagram (13) commutes.

We are mainly interested in the case where A is the source P of a Dirac morphismfor H . Then K�C1.EX=BX / D Ktop�

�H; cred

H .X;D/�, and the top row in (13) is the

H -equivariant coarse co-assembly map forX with coefficientsD. Since we assumeHto act properly onX , we have a KKG-equivalenceBX � C0.X;D/ÌH , and similarlyfor P .X/. Hence we now get a commuting square

Ktop�C1

�H; cred

H .X;D/� @ ��

��

KX�H .X;D/

�

��

KKG� .C; C0.X;D/ ÌH/p�

EG �� RKKG� .EGIC; C0.X;D/ ÌH/.

(14)


If, in addition, D D C and the action of H on X is free, then we can further simplifythis to

Ktop�C1

�H; cred

H .X/� @ ��

��

KX�H .X/

�

��

KKG��C; C0.X=H/

� p�

EG �� RKKG��EGIC; C0.X=H/

�.

(15)

We may also specialise the space X to jGj, with G acting by multiplication on theleft, and with H � G a compact subgroup acting on jGj by right multiplication. Thisis the special case of (13) that is used in [7]. The following applications will requireother choices of X .

3.1 Applications to Lipschitz classes. Now we use Lemma 7 to construct interestingelements in KKG�

�C; C0.X/

�for aG-spaceX . This is related to the method of Lipschitz

maps developed by Connes, Gromov and Moscovici in [3].

3.1.1 Pulled-back coarse structures. Let X be a G-space, let Y be a coarse spaceand let ˛ W X ! Y be a proper continuous map. We pull back the coarse structureon Y to a coarse structure on X , letting E � X � X be an entourage if and only if˛�.E/ � Y � Y is one. Since ˛ is proper and continuous, this coarse structure iscompatible with the topology on X . For this coarse structure, G acts by translationsif and only if ˛ satisfies the following displacement condition used in [3]: for anycompact subset K � G, the set

˚�˛.gx/; ˛.x/

� 2 Y � Y j x 2 X; g 2 K�is an entourage of Y . The map ˛ becomes a coarse map. Hence we obtain a commutingdiagram

K�C1�cred.Y /

� ˛�

��

@Y

��

K�C1�cred.X/

� ��

@X

��

KKG��C; C0.X/

�p�

EG

��

KX�.Y / ˛�

�� KX�.X/ �� RKKG�

�EGIC; C0.X/

�:

with and � as in Lemma 7.The constructions of [3, §I.10] only use Y D RN with the Euclidean coarse struc-

ture. The coarse co-assembly map is an isomorphism for RN because RN is scalable.Moreover, RN is uniformly contractible and has bounded geometry. Hence we obtaincanonical isomorphisms

K�C1�cred.RN /

� Š KX�.RN / Š K�.RN /:

In particular, K�C1�cred.RN /

� Š Z with generator Œ@RN � in K1�N�cred.RN /

�. This

class is nothing but the usual dual-Dirac morphism for the locally compact group RN .


As a result, any map ˛ W X ! RN that satisfies the displacement condition aboveinduces

Œ˛� WD �˛�Œ@RN �

� 2 KKG�N�C; C0.X/

�:

The commutative diagram (13) computesp�EGŒ˛� 2 RKKG�N

�EGIC; C0.X/

�in purely

topological terms.

3.1.2 Principal bundles over coarse spaces. As in [3], we may replace a fixed mapX ! RN by a section of a vector bundle over X . But we need this bundle tohave a G-equivariant spin structure. To encode this, we consider a G-equivariantSpin.N /-principal bundle � W E ! B together with actions of G on E and B suchthat� isG-equivariant and the action onE commutes with the action ofH WD Spin.N /.Let T WD E �Spin.N/ RN be the associated vector bundle over B . It carries a G-in-variant Euclidean metric and spin structure. As is well-known, sections ˛ W B ! T

correspond bijectively to Spin.N /-equivariant maps ˛0 W E ! RN ; here a section ˛corresponds to the map ˛0 W E ! RN that sends y 2 E to the coordinates of ˛�.y/ inthe orthogonal frame described by y. Since the group Spin.N / is compact, the map ˛0is proper if and only if b 7! k˛.b/k is a proper function on B .

As in §3.1.1, a Spin.N /-equivariant proper continuous map ˛0 W E ! Y for a coarsespace Y allows us to pull back the coarse structure of Y to E; then Spin.N / acts byisometries. The groupG acts by translations if and only if ˛0 satisfies the displacementcondition from §3.1.1. If Y D RN , we can rewrite this in terms of ˛ W B ! T : weneed

supfkg˛.g�1b/ � ˛.b/kjb 2 B; g 2 Kgto be bounded for all compact subsets K � G.

If the displacement condition holds, then we are in the situation of Lemma 7 withH D Spin.N / and X D E. Since H acts freely on E, C0.E/ÌH is G-equivariantlyMorita–Rieffel equivalent to C0.B/. We obtain canonical maps

KSpin.N/�C1

�cred

Spin.N/.RN /

� .˛0/��! KSpin.N/�C1

�cred

Spin.N/.E/�

�! KKG��C; C0.E/ Ì Spin.N /

� Š KKG��C; C0.B/

�:

The Spin.N /-equivariant coarse co-assembly map for RN is an isomorphism by [7]because the group RN Ì Spin.N / has a dual-Dirac morphism. Using also the uniformcontractibility of RN and Spin.N /-equivariant Bott periodicity, we get

KSpin.N/�C1

�cred

Spin.N/.RN /

� Š KX�Spin.N/.RN / Š K�Spin.N/.R

N / Š K�CNSpin.N/.point/:

The class of the trivial representation in Rep.SpinN/ Š KSpin.N/0 .C/ is mapped to the

usual dual-Dirac morphism Œ@RN � 2 KSpin.N/1�N

�cred

Spin.N/.RN /

�for RN . As a result, any

proper section ˛ W B ! T satisfying the displacement condition induces

Œ˛� WD �˛�Œ@RN �

� 2 KKG�N�C; C0.B/

�:


Again, the commutative diagram (13) computes p�EGŒ˛� 2 RKKG�N

�EGIC; C0.X/

�in purely topological terms.

3.1.3 Coarse structures on jet bundles. LetM be an oriented compact manifold andlet DiffC.M/ be the infinite-dimensional Lie group of orientation-preserving diffeo-morphisms of M . Let G be a locally compact group that acts on M by a continuousgroup homomorphism G ! DiffC.M/. The Gelfand–Fuchs cohomology of M ispart of the group cohomology of DiffC.M/ and by functoriality maps to the groupcohomology of G. It is shown in [3] that the range of Gelfand–Fuchs cohomology inthe cohomology of G yields homotopy-invariant higher signatures.

This argument has two parts; one is geometric and concerns the construction of aclass in KKG�

�C; C0.X/

�for a suitable space X ; the other uses cyclic homology to

construct linear functionals on K�.C0.X/ Ìr G/ associated to Gelfand–Fuchs coho-mology classes. We can simplify the first step; the second has nothing to do with coarsegeometry.

Let �k W JkC.M/ ! M be the oriented k-jet bundle over M . That is, a point inJkC.M/ is the kth order Taylor series at 0 of an orientation-preserving diffeomorphismfrom a neighbourhood of 0 2 Rn into M . This is a principal H -bundle over M ,whereH is a connected Lie group whose Lie algebra h is the space of polynomial mapsp W Rn ! Rn of order k with p.0/ D 0, with an appropriate Lie algebra structure.The maximal compact subgroup K � H is isomorphic to SO.n/, acting by isometrieson Rn. It acts on h by conjugation.

Since our construction is natural, the action of G onM lifts to an action on JkC.M/

that commutes with the H -action. We let H act on the right and G on the left. DefineXk WD JkC.M/=K. This is the bundle space of a fibration over M with fibres H=K.Gelfand–Fuchs cohomology can be computed using a chain complex of DiffC.M/-invariant differential forms on Xk for k ! 1. Using this description, Connes,Gromov, and Moscovici associate to a Gelfand–Fuchs cohomology class a functionalK�.C0.Xk/ Ìr G/! C for sufficiently high k in [3].

Since JkC.M/=H ŠM is compact, there is a unique coarse structure on JkC.M/ forwhich H acts isometrically (see §2.2). With this coarse structure, JkC.M/ is coarselyequivalent to H . The compactness of JkC.M/=H Š M also implies easily that Gacts by translations. We have a Morita–Rieffel equivalence C0.Xk/ � C0.JkCM/ ÌKbecause K acts freely on JkC.M/. We want to study the map

K�C1.credK .JkCM/ ÌK/

�! KKG� .C; C0.JkCM/ ÌK/ Š KKG��C; C0.Xk/

�

produced by Lemma 7.Since H is almost connected, it has a dual-Dirac morphism by [10]; hence the

K-equivariant coarse co-assembly map for H is an isomorphism by the main resultof [7]. Moreover, H=K is a model for EG by [1]. We get

K�C1.credK .JkCM/ ÌK/ Š K�C1.cred

K .jH j/ ÌK/ Š KX�K.jH j/ Š K�K.H=K/:


Let h and k be the Lie algebras ofH andK. There is aK-equivariant homeomorphismh=k Š H=K, where K acts on h=k by conjugation. Now we need to know whetherthere is a K-equivariant spin structure on h=k. One can check that this is the case ifk 0; 1 mod 4. Since we can choose k as large as we like, we can always assumethat this is the case. The spin structure allows us to use Bott periodicity to identifyK�K.H=K/ Š KH��N .C/, which is the representation ring of K in degree �N , whereN D dim h=k. Using our construction, the trivial representation ofK yields a canonicalelement in KKG�N

�C; C0.Xk/

�.

This construction is much shorter than the corresponding one in [3] because we useKasparov’s result about dual-Dirac morphisms for almost connected groups. Much ofthe corresponding argument in [3] is concerned with proving a variation on this resultof Kasparov.

4 Computation of KKG� .C; P/

So far we have merely used the diagram (13) to construct certain elements in KKG� .C;B/.Now we show that this construction yields an isomorphic description of KKG� .C;P/.This assertion requiresG to be a totally disconnected group with aG-compact universalproper G-space. We assume this throughout this section.

Lemma 8. In the situation of Lemma 7, suppose that X D jGj with G acting by lefttranslations and thatH � G is a compact subgroup acting onX by right translations;here jGj carries the coarse structure of Example 3. Then the maps and � areisomorphisms.

Proof. We reduce this assertion to results of [7]. The C�-algebras credH .jGj;D/ Ì H

and credH .jGj;D/H are strongly Morita equivalent, whence have isomorphic K-theory.

It is shown in [7] that

K�C1�credH .jGj;D/H � Š KKG� .C; IndGH D/: (16)

Finally, IndGH .D/ D C0.G;D/H is G-equivariantly Morita–Rieffel equivalent to

C0.G;D/ ÌH . Hence we get

K�C1.credH .jGj;D/ ÌH/ Š K�C1

�credH .jGj;D/H � Š KKG�

�C; IndGH .D/

�Š KKG�

�C; C0.G;D/ ÌH

�:

(17)

It is a routine exercise to verify that this composition agrees with the map in (13).Similar considerations apply to the map �.

We now setX D jGj, and letG D H . The actions ofG on jGj on the left and rightare by translations and isometries, respectively. Lemma 7 yields a map

‰� W K�C1�.cred.jGj;D/˝max A/ÌG

�! KKG��C; C0.jGj;D˝max A/ÌG

�: (18)


for all A;D, where we use the G-equivariant Morita-Rieffel equivalence betweenC0.jGj;D/ ÌG and D. It fits into a commuting diagram

K�C1�.credG .jGj;D/˝max A/ ÌG

� @ ��

‰D;A�

��

K�.C0.EG;D ˝max A/ ÌG/

Š��

KKG� .C;D ˝max A/p�

EG �� RKKG� .EGIC;D ˝max A/.

Lemma 9. The class ofG-C�-algebrasA for which‰D;A� is an isomorphism for allDis triangulated and thick and contains all G-C�-algebras of the form C0.G=H/ forcompact open subgroups H of G.

Proof. The fact that this category of algebras is triangulated and thick means that it isclosed under suspensions, extensions, and direct summands. These formal propertiesare easy to check.

Since H � G is open, there is no difference between H -continuity and G-conti-nuity. Hence

�credG .jGj;D/˝max C0.G=H/

�ÌG Š �

credH .jGj;D/˝max C0.G=H/

�ÌG

� �credH .jGj;D/ ÌH;

where � means Morita-Rieffel equivalence. Similar simplifications can be made inother corners of the square. Hence the diagram forA D C0.G=H/ andG acting on theright is equivalent to a corresponding diagram for trivial A and H acting on the right.The latter case is contained in Lemma 8.

Theorem 10. Let G be an almost totally disconnected group with G-compact EG.Then for every B 2 KKG , the map

‰� W Ktop�C1

�G; cred.jGj; B/�! KKG� .C; B ˝ P/

is an isomorphism and the diagram

Ktop�C1

�G; cred.jGj; B/� ��

��

Š ‰�

��

KX�G.jGj; B/Š��

KKG� .C; B ˝ P/p�

EG �� RKKG� .EGIC; B ˝ P/

commutes. In particular, Ktop�C1

�G; cred.jGj/� is naturally isomorphic to KKG� .C;P/.

Proof. It is shown in [7] that for such groupsG, the algebra P belongs to the thick trian-gulated subcategory of KKG that is generated by C0.G=H/ for compact subgroupsHof G. Hence the assertion follows from Lemma 9 and our definition of Ktop.


Corollary 11. Let D 2 KKG.P;C/ be a Dirac morphism for G. Then the followingdiagram commutes:

Ktop�C1

�G; cred

G .jGj/��

��

‰�Š��

@ �� K�.C0.EG;P/ ÌG/

Š��

D�

Š�� K�.C0.EG/ ÌG/

Š��

KKG� .C;P/p�

EG ��

D� ��RKKG� .EGIC;P/

D�

Š�� RKKG� .EGIC;C/

KKG� .C;C/,p�

EG

��

where ‰� is as in Theorem 10 and �� is the G-equivariant coarse co-assembly mapfor jGj, and the indicated maps are isomorphisms.

Proof. This follows from Theorem 10 and the general properties of the Dirac morphismdiscussed in §2.1.

Definition 12. We call a 2 RKKG� .EGIC;C/(a) a boundary class if it lies in the range of

�� W Ktop�C1

�G; cred.jGj/�! RKKG� .EGIC;C/I

(b) properly factorisable if a D p�EG.b ˝A c/ for some proper G-C�-algebra A and

some b 2 KKG� .C; A/, c 2 KKG� .A;C/;

(c) proper Lipschitz if a D p�EG.b ˝C0.X/ c/, where b 2 KKG�

�C; C0.X// is con-

structed as in §3.1.1 and §3.1.2 and c 2 KKG� .C0.X/;C/ is arbitrary.

Proposition 13. Let G be a totally disconnected group with G-compact EG.

(a) A classa 2 RKKG� .EGIC;C/ is properly factorisable if and only if it is a boundaryclass.

(b) Proper Lipschitz classes are boundary classes.

(c) The boundary classes form an ideal in the ring RKKG� .EGIC;C/.(d) The class 1EG is a boundary class if and only if G has a dual-Dirac morphism; in

this case, the G-equivariant coarse co-assembly map �� is an isomorphism.

(e) Any boundary class lies in the range of

p�EG W KKG� .C;C/! RKKG� .EGIC;C/and hence yields homotopy invariants for manifolds.


Proof. By Corollary 11, the equivariant coarse co-assembly map

�� W Ktop�C1

�G; cred.jGj/�! K�

�C0.EG ÌG/

� Š RKKG� .EGIC;C/is equivalent to the map

KKG� .C;P/p�

EG��! RKKG� .EGIC;P/ Š RKKG� .EGIC;C/:If we combine this with the isomorphism RKKG� .EGIC;C/ Š KKG� .P;P/, the result-ing map

KKG� .C;P/! KKG� .P;P/

is simply the product (on the left) with D 2 KKG.P;C/; this map is known to be anisomorphism if and only if it is surjective, if and only if 1P is in its range, if and onlyif the H -equivariant coarse co-assembly map

K�.credH .jGj/ ÌH/! KX�H .jGj/

is an isomorphism for all compact subgroups H of G by [7].For any G-C�-algebra B , the Z=2-graded group Ktop� .G;B/ Š K�

�.B ˝ P/Ìr G

�is a graded module over the Z=2-graded ring KKG� .P;P/ Š RKKG� .EGIC;C/ ina canonical way; the isomorphism between these two groups is a ring isomorphismbecause it is the composite of the two ring isomorphisms

KKG� .P;P/p�

EG��!Š RKKG� .EGIP;P/˝P ��Š RKKG� .EGIC;C/:

Hence we get module structures on Ktop��G; cred.jGj/� and

Ktop��G;C0.EG;K/

� Š K�.C0.EG/ ÌG/ Š RKKG� .EGIC;C/:

The latter isomorphism is a module isomorphism; thus Ktop��G;C0.EG;K/

�is a free

module of rank1. The equivariant co-assembly map is natural in the formal sense, so thatit is an RKKG� .EGIC;C/-module homomorphism. Hence its range is a submodule,that is, an ideal in RKKG� .EGIC;C/ (since this ring is graded commutative, there isno difference between one- and two-sided graded ideals). This also yields that �� issurjective if and only if it is bijective, if and only if the unit class 1EG belongs to itsrange; we already know this from [7].

If A is a properG-C�-algebra, then idA˝D 2 KKG.A˝P; A/ is invertible ([11]).If b 2 KKG� .C; A/ and c 2 KKG� .A;C/, then we can write the Kasparov productb ˝A c as

Cb�! A

idA˝D ��Š A˝ Pc˝idP��! P

D�! C;

where the arrows are morphisms in the category KKG . Therefore, b ˝A c factorsthrough D and hence is a boundary class by Theorem 10.


5 Dual-Dirac morphisms and the Carlsson–Pedersen condition

Now we construct boundary classes in RKKG� .EGIC;C/ from more classical bound-aries. We suppose again that EG is G-compact, so that EG is a coarse space.

A metrisable compactification of EG is a metrisable compact spaceZ with a home-omorphism between EG and a dense open subset of Z. It is called coarse if allscalar-valued functions on Z have vanishing variation; this implies the correspondingassertion for operator-valued functions because C.Z;D/ Š C.Z/˝D. Equivalently,the embedding EG ! Z factors through the Higson compactification of EG. A com-pactification is called G-equivariant if Z is a G-space and the embedding EG ! Z

is G-equivariant. An equivariant compactification is called strongly contractible if itis H -equivariantly contractible for all compact subgroups H of G. The Carlsson–Pe-dersen condition requires that there should be a G-compact model for EG that has acoarse, strongly contractible, and G-equivariant compactification.

Typical examples of such compactifications are the Gromov boundary for a hy-perbolic group (viewed as a compactification of the Rips complex), or the visibilityboundary of a CAT(0) space on whichG acts properly, isometrically, and cocompactly.

Theorem 14. Let G be a locally compact group with a G-compact model for EG

and let EG � Z be a coarse, strongly contractible, G-equivariant compactification.Then G has a dual-Dirac morphism.

Proof. We use the C�-algebra xBredG .Z/ as defined in §2.3. Since Z is coarse, there

is an embedding xBredG .Z/ � Ncred

G .EG/. Let @Z WD Z n EG be the boundary of thecompactification. Identifying

xBredG .@Z/ Š xBred

G .Z/=C0.EG;KG/;

we obtain a morphism of extensions

0 �� C0.EG;KG/ �� NcredG .EG/ �� cred

G .EG/ �� 0

0 �� C0.EG;KG/ �� xBredG .Z/

��

�� xBredG .@Z/

��

�� 0.

Let H be a compact subgroup. Since Z is compact, we have xBG.Z/ D C.Z;K/.Since Z is H -equivariantly contractible by hypothesis, xBG.Z/ is H -equivariantlyhomotopy equivalent to C. Hence xBred

G .Z/ has vanishing H -equivariant K-theory.This implies Ktop�

�G; xBred

G .Z/� D 0 by [2], so that the connecting map

Ktop�C1

�G; xBred

G .@Z/�! Ktop�

�G;C0.EG/

� Š K�.C0.EG/ ÌG/ (19)

is an isomorphism. This in turn implies that the connecting map

Ktop�C1

�G; cred

G .EG/�! Ktop�

�G;C0.EG/

�


is surjective. Thus we can lift 1 2 RKKG0 .EGIC;C/ Š K0.C0.EG/ ÌG/ to

˛ 2 Ktop1

�G; cred

G .EG/� Š Ktop

1

�G; cred

G .jGj/�:Then ‰�.˛/ 2 KKG0 .C;P/ is the desired dual-Dirac morphism.

The group Ktop��G; xBred

G .@Z/�

that appears in the above argument is a reduced

topological G-equivariant K-theory for @Z and hence differs from Ktop��G;C.@Z/

�.

The relationship between these two groups is analysed in [6].

References

[1] H. Abels, Parallelizability of proper actions, global K-slices and maximal compact sub-groups, Math. Ann. 212 (1974/75), 1–19.

[2] J. Chabert, S. Echterhoff, H. Oyono-Oyono, Going-down functors, the Künneth Formula,and the Baum-Connes conjecture, Geom. Funct. Anal. 14 (2004), 491–528.

[3] A. Connes, A., M. Gromov, H. Moscovici, H. Group cohomology with Lipschitz controland higher signatures, Geom. Funct. Anal. 3 (1993), 1–78.

[4] A. N. Dranishnikov, Lipschitz cohomology, Novikov’s conjecture, and expanders, Tr. Mat.Inst. Steklova 247 (2004), 59–73. translation: Proc. Steklov Inst. Math. 4 (247) (2004),50–63.

[5] H. Emerson, R. Meyer, Dualizing the coarse assembly map, J. Inst. Math. Jussieu 5 (2006),161–186.

[6] H. Emerson, R. Meyer, Euler characteristics and Gysin sequences for group actions onboundaries, Math. Ann. 334 (2006), 853–904.

[7] H. Emerson, R. Meyer, A descent principle for the Dirac–dual-Dirac method, Topology 46(2007), 185–209.

[8] S. C. Ferry, A. Ranicki, J. Rosenberg, Novikov conjectures, index theorems and rigidity.Vol. 1, London Math. Soc. Lecture Notes Ser. 226, Cambridge University Press, Cambridge1995.

[9] N. Higson, BivariantK-theory and the Novikov conjecture, Geom. Funct. Anal. 10 (2000),563–581.

[10] G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91(1988), 147–201.

[11] R. Meyer, R. Nest, The Baum–Connes conjecture via localisation of categories, Topology45 (2006), 209–259.

On K1 of a Waldhausen category

Fernando Muro and Andrew Tonks�

Introduction

A general notion ofK-theory, for a category W with cofibrations and weak equivalences,was defined by Waldhausen [10] as the homotopy groups of the loop space of a certainsimplicial category wS .W,

KnW D �n� jwS .Wj Š �nC1 jwS .Wj ; n � 0:Waldhausen K-theory generalizes the K-theory of an exact category E, defined as thehomotopy groups of � jQEj, where QE is a category defined by Quillen.

Gillet and Grayson defined in [2] a simplicial setG.E which is a model for� jQEj.This allows one to compute K1E as a fundamental group, K1E D �1 jG.Ej. Usingthe standard techniques for computing �1 Gillet and Grayson produced algebraic rep-resentatives for arbitrary elements in K1E. These representatives were simplified bySherman [8], [9] and further simplified by Nenashev [5]. Nenashev’s representativesare pairs of short exact sequences on the same objects,

A�� B

�� C:

Such a pair gives a loop in jG.Ej which corresponds to a 2-sphere in jwS .Ej, obtainedby pasting the 2-simplices associated to each short exact sequence along their commonboundary.

B

A

��

��

�

��

��

� C��

��

��

��

��

��

Figure 1. Nenashev’s representative of an element in K1E.

Nenashev showed in [6] certain relations which are satisfied by these pairs of shortexact sequences, and he later proved in [7] that pairs of short exact sequences togetherwith these relations yield a presentation of K1E.

�The first author was partially supported by the MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the MEC postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research cantract; and thesecond author by the MEC-FEDER grant MTM2004-03629.

92 F. Muro and A. Tonks

In this paper we produce representatives for all elements in K1 of an arbitraryWaldhausen category W which are as close as possible to Nenashev’s representativesfor exact categories. They are given by pairs of cofiber sequences where the cofibrationshave the same source and target. The cofibers, however, may be non-isomorphic, butthey are weakly equivalent via a length 2 zig-zag.

C1 �� A

�� B

��

�� C:

C2 ��

This diagram is called a pair of weak cofiber sequences. It corresponds to a 2-sphere injwS .Wj obtained by pasting the 2-simplices associated to the cofiber sequences alongthe common part of the boundary. The two edges which remain free after this operationare filled with a disk made of two pieces corresponding to the weak equivalences.

B

A

��

��

�

��

��

�

C1

C2

C��

��

��

��

��

�

��

Figure 2. Our representative of an element in K1W.

In addition we show in Section 3 that pairs of weak cofiber sequences satisfy acertain generalization of Nenashev’s relations.

Let Kwcs1 W be the group generated by the pairs of weak cofiber sequences in W

modulo the relations in Section 3. The results of this paper show in particular that thereis a natural surjection

Kwcs1 W � K1W

which, by [7], is an isomorphism in case W D E is an exact category. See Section 6for a comparison of Kwcs

1 E with Nenashev’s presentation. We do not know whetherthis natural surjection is an isomorphism for an arbitrary Waldhausen category W, butwe prove here further evidence which supports the conjecture: the homomorphism

� � W K0W˝ Z=2! K1W

induced by the action of the Hopf map � 2 �s� in the stable homotopy groups of spheresfactors as

K0W˝ Z=2 �! Kwcs1 W � K1W:

We finish this introduction with some comments about the method of proof. We donot rely on any previous similar result because we do not know of any generalization of

On K1 of a Waldhausen category 93

the Gillet–Grayson construction for an arbitrary Waldhausen category W. The largestknown class over which the Gillet–Grayson construction works is the class of pseudo-additive categories, which extends the class of exact categories but still does not coverall Waldhausen categories, see [3]. We use instead the algebraic model D�W definedin [4] for the 1-type of the Waldhausen K-theory spectrum KW. The algebraic objectD�W is a chain complex of non-abelian groups concentrated in dimensions i D 0; 1

whose homology is KiW,

.D0W/ab ˝ .D0W/ab

h � ;� i��

K1W � � �� D1W @ �� D0W �� K0W;

(A)

i.e. the lower row is exact. This non-abelian chain complex is equipped with a bilinearmap h � ; � i which determines the commutators. This makes D�W a stable quadraticmodule in the sense of [1]. This stable quadratic module is defined in [4] in terms ofgenerators and relations. Generators correspond simply to objects, weak equivalences,and cofiber sequences in W.

Acknowledgement. The authors are very grateful to Grigory Garkusha for askingabout the relation between the algebraic model for WaldhausenK-theory defined in [4]and Nenashev’s presentation of K1 of an exact category in [7].

1 The algebraic model for K0 and K1

Definition 1.1. A stable quadratic module C� is a diagram of group homomorphisms

C ab0 ˝ C ab

0

h � ;� i��! C1@�! C0

such that given ci ; di 2 Ci , i D 0; 1,

1. @hc0; d0i D Œd0; c0�,2. [email protected]/; @.d1/i D Œd1; c1�,3. hc0; d0i C hd0; c0i D 0.

Here Œx; y� D �x � y C x C y is the commutator of two elements x; y 2 K in anygroupK, andKab is the abelianization ofK. It follows from the axioms that the imageof h � ; � i and Ker @ are central in C1, the groups C0 and C1 have nilpotency class 2, [email protected]/ is a normal subgroup of C0.

A morphism f W C� ! D� of stable quadratic modules is given by group homo-morphisms fi W Ci ! Di , i D 0; 1, compatible with the structure homomorphisms ofC� and D�, i.e. f0@ D @f1 and f1h � ; � i D hf0; f0i.

Stable quadratic modules were introduced in [1, Definition IV.C.1]. Notice, how-ever, that we adopt the opposite convention for the homomorphism h � ; � i, i.e. we com-pose with the symmetry isomorphism on the tensor square.


Remark 1.2. There is a natural right action of C0 on C1 defined by

cc01 D c1 C hc0; @.c1/i:

The axioms of a stable quadratic module imply that commutators in C0 act trivially onC1, and that C0 acts trivially on the image of h � ; � i and on Ker @.

The action gives @ W C1 ! C0 the structure of a crossed module. Indeed a stablequadratic module is the same as a commutative monoid in the category of crossedmodules such that the monoid product of two elements in C0 vanishes when one ofthem is a commutator, see [4, Lemma 4.18].

A stable quadratic module can be defined by generators and relations in degree 0and 1. In the appendix we give details about the construction of a stable quadraticmodule defined by generators and relations. This is useful to understand Definition 1.3below from a purely group-theoretic perspective.

We assume the reader has certain familiarity with Waldhausen categories and relatedconcepts. We refer to [12] for the basics, see also [11]. The following definition wasintroduced in [4].

Definition 1.3. Let C be a Waldhausen category with distinguished zero object �, and

cofibrations and weak equivalences denoted by � and�!, respectively. A generic

cofiber sequence is denoted by

A � B � B=A:

We define D�C as the stable quadratic module generated in dimension zero by thesymbols

• ŒA� for any object in C,

and in dimension one by

• ŒA�! A0� for any weak equivalence,

• ŒA � B � B=A� for any cofiber sequence,

such that the following relations hold.

(R1) @ŒA�! A0� D �ŒA0�C ŒA�.

(R2) @ŒA � B � B=A� D �ŒB�C ŒB=A�C ŒA�.(R3) Œ�� D 0.

(R4) ŒA1A�! A� D 0.

(R5) ŒA1A�! A � �� D 0, Œ�� A

1A�! A� D 0.


(R6) For any pair of composable weak equivalences A�! B

�! C ,

ŒA�! C � D ŒB �! C �C ŒA �! B�:

(R7) For any commutative diagram in C as follows

A ��

��

B ��

��

B=A

��

A0 �� B 0 �� B 0=A0

we have

ŒA�! A0�C ŒB=A �! B 0=A0�ŒA� D� ŒA0 � B 0 � B 0=A0�

C ŒB �! B 0�C ŒA � B � B=A�:

(R8) For any commutative diagram consisting of four cofiber sequences in C as follows

C=B

B=A �� C=A

��

A �� B ��

��

C

��

we have

ŒB � C � C=B�C ŒA � B � B=A�

D ŒA � C � C=A�C ŒB=A � C=A � C=B�ŒA�:

(R9) For any pair of objects A;B in C

hŒA�; ŒB�i D �ŒB i2� A _ B p1� A�C ŒA i1� A _ B p2� B�:

Here

Ai1 ��

A _ Bp1

��p2

�� Bi2��

are the inclusions and projections of a coproduct in C.

Remark 1.4. These relations are quite natural, and some illustration of their meaningis given in [4, Figure 2]. They are not however minimal: relation (R3) follows from(R2) and (R5), and (R4) follows from (R6). Also (R5) is equivalent to


(R50) Œ�� D 0.

This equivalence follows from (R8) on considering the diagrams

�

� ��

��

A ��1A �� A ��

1A ��

��

A,

��

A

� �� A

1A

��

� ��

��

A.

1A

��

We now introduce the basic object of study of this paper.

Definition 1.5. A weak cofiber sequence in a Waldhausen category C is a diagram

A � B � C1� C

given by a cofiber sequence followed by a weak equivalence in the opposite direction.

We associate the element

ŒA � B � C1� C � D ŒA � B � C1�C ŒC �! C1�

ŒA� 2 D�C (1.1)

to any weak cofiber sequence. By (R1) and (R2) we have

@ŒA � B � C1� C � D �ŒB�C ŒC �C ŒA�:

The following fundamental identity for weak cofiber sequences is a slightly lesstrivial consequence of the relations (R1)–(R9).

Proposition 1.6. Consider a commutative diagram

A0 ��jA

��

j 0

��

ArA ��

��

j

��

A000��

j 000

��

A00��

j 00

��

�wA

��

B 0 ��jB

��

r 0

��

BrB ��

r

��

B 000

r 000

��

B 00

r 00

��

�wB

��

D0 ��jD

�� DrD �� D000 D00�

wD��

C 0 ��jC

��

�w0

��

CrC ��

�w

��

C 000

�w000

��

C 00

�w00

��

�wC

��


such that the induced map j [A0 jB W A [A0 B 0 � B is a cofibration. Denote thehorizontal weak cofiber sequences by lA, lB , lD and lC , and the vertical ones by l 0, l ,l 000 and l 00. Then the following equation holds in D�C.

�ŒlA� � ŒlC �ŒA� � Œl �C ŒlB �C Œl 00�ŒB0� C Œl 0� D hŒC 0�; ŒA00�i:

Proof. Applying (R7) to the two obvious equivalences of cofiber sequences gives

ŒD0 � D � D000� D Œw�C ŒlC � � ŒwC �ŒC 0� � Œw000�ŒC 0� � Œw0�; (a)

ŒA000 � B 000 � D000� D ŒwB �C Œl 00� � Œw00�ŒA00� � ŒwD�ŒA00� � ŒwA�; (b)

using the notation of (1.1).

Now consider the following commutative diagrams:

A0 ��jA

��

j 0

��

push

A��

��

��

j

��

B 0 ��

jB ��

X ��

��

��

��

B ,

A0 ��jA

��

j 0

��

push

A��

��i1r

A

��

B 0 ��

i2r0 ��

X

��

��

��

A000 _D0.

We now have four commutative diagrams of cofiber sequences as in (R8).

c) D0

A000 �� A000 _D0

��

A0 �� A ��

��

X

��

d) A000

D0 �� A000 _D0

��

A0 �� B 0 ��

��

X

��

e) D000

D0 �� D

��

A �� X ��

��

B

��

f) D000

A000 �� B 000

��

B 0 �� X ��

��

B

��


Therefore we obtain the corresponding relations

ŒA � X � D0�C ŒlA� � ŒwA�ŒA0� (c)

D ŒA0 � X � A000 _D0�C ŒA000 � A000 _D0 � D0�ŒA0�;

ŒB 0 � X � A000�C Œl 0� � Œw0�ŒA0� (d)

D ŒA0 � X � A000 _D0�C ŒD0 � A000 _D0 � A000�ŒA0�;

ŒX � B � D000�C ŒA � X � D0� (e)

D Œl � � Œw�ŒA� C ŒD0 � D � D000�ŒA�

D Œl �C .ŒlC � � ŒwC �ŒC 0� � Œw000�ŒC 0� � Œw0�/ŒA�;ŒX � B � D000�C ŒB 0 � X � A000� (f)

D ŒlB � � ŒwB �ŒB0� C ŒA000 � B 000 � D000�ŒB0�

D ŒlB �C .Œl 00� � Œw00�ŒA00� � ŒwD�ŒA00� � ŒwA�/ŒB0�:

Here we have used equations (a) and (b). Now �.c/C .d/ and �.e/C .f/ yield

ŒwA�ŒA0� � ŒlA� � ŒA � X � D0�C ŒB 0 � X � A000�C Œl 0� � Œw0�ŒA0� (g)

D �ŒA000 � A000 _D0 � D0�ŒA0� C ŒD0 � A000 _D0 � A000�ŒA0�

D hŒD0�; ŒA000�i;� ŒA � X � D0�C ŒB 0 � X � A000� (h)

D Œw0�ŒA� C aŒC 0�CŒA� � ŒlC �ŒA� � Œl �C ŒlB �C Œl 00�ŒB0� � aŒA00�CŒB0� � ŒwA�ŒB0�;

where we use (R9) and Remark 1.2, and we write a D Œw000�C ŒwC � D ŒwD�C Œw00�,by (R6). In fact it is easy to check by using Remark 1.2 and the laws of stable quadraticmodules that the terms aŒC

0�CŒA� and aŒA00�CŒB0� cancel in this expression, since

@� � ŒlC �ŒA� � Œl �C ŒlB �C Œl 00�ŒB0�

� D �.ŒC 0�C ŒA�/C .ŒA00�C ŒB 0�/:Substituting (h) into the left hand side of (g) we obtain

ŒwA�ŒA0� � ŒlA�C Œw0�ŒA� � ŒlC �ŒA� � Œl �C ŒlB �C Œl 00�ŒB0� � ŒwA�ŒB0� C Œl 0� � Œw0�ŒA0�

D hŒD0�; ŒA000�i;which, using again Definition 1.1 and Remark 1.2, may be rewritten as

� ŒlA� � ŒlC �ŒA� � Œl �C ŒlB �C Œl 00�ŒB0� C Œl 0�D � Œw0�ŒA00�CŒA0� � ŒwA�ŒA0� � hŒA000�; ŒD0�i C Œw0�ŒA0� C ŒwA�ŒC 0�CŒA0�:

The result then follows from the identity

h@ŒwA�; @Œw0�i � hŒA00�; @Œw0�i C hŒC 0�; @ŒwA�i D hŒC 0�; ŒA00�i C hŒA000�; ŒD0�i:


2 Pairs of weak cofiber sequences and K1

Let C be a Waldhausen category. A pair of weak cofiber sequences is a diagram givenby two weak cofiber sequences with the first, second, and fourth objects in common:

C1 �� A

�� B

��

�� C

C2 . ��

We associate to any such pair the following element

8<ˆ:

C1 �� w1

��

A��j1 ��j2

�� B

r1 ��

r2 ��

� C

C2 �

w2 ��

9>>=>>;D �ŒA j1� B

r1� C1� w1C �C ŒA j2� B

r2� C2� w2C �

D ŒA j2� Br2� C2

� w2C � � ŒA j1� B

r1� C1� w1C �;

(2.1)

which lies in K1C D Ker @, see diagram (A) in the introduction. For the secondequality in (2.1) we use that Ker @ is central, see Remark 1.2, so we can permute theterms cyclically.

The following theorem is one of the main results of this paper.

Theorem 2.1. Any element inK1C is represented by a pair of weak cofiber sequences.

This theorem will be proved later in this paper. We first give a set of useful relationsbetween pairs of weak cofiber sequences and develop a sum-normalized version of themodel D�C.

3 Relations between pairs of weak cofiber sequences

Suppose we have six pairs of weak cofiber sequences in a Waldhausen category C

A001 ��

�wA1��

A0 ��jA1 ��

jA2

�� A

rA1 ��

rA2

�� A00;

A002�wA2

��

B 001 ��

�wB1��

B 0 ��jB1 ��

jB2

�� B

rB1 ��

rB2

�� B 00;

B 002�wB2

��

LC 001 ��

�wC1��

C 0 ��jC1 ��

jC2

�� C

rC1 ��

rC2

��

C 00;

LC 002�wC2

��

C 01 ��

�w0

1��

A0 ��j 0

1 ��

j 0

2

�� B 0r 0

1 ��

r 0

2

��

C 0;

C 02��w0

2

��

C1 ��

�w1��

A��j1 ��j2

�� B

r1 ��

r2

C;

C2�w2

��

OC 001 ��

�w00

1��

A00 ��j 00

1 ��

j 00

2

�� B 00r 00

1 ��

r 00

2

C 00;

OC 002�w00

2

��


that we denote for the sake of simplicity as �A, �B , �C , �0, �, and �00, respectively.Moreover, assume that for i D 1; 2 there exists a commutative diagram

A0 ��jAi ��

��

j 0

i

��

A��

ji

��

rAi �� A00i��

L|i��

A00��

j 00

i

��

�wAi

��

B 0 ��jBi ��

r 0

i

��

BrBi ��

ri

��

B 00i

Lri��

B 00

r 00

i

��

�wBi

��

C 0i ��O|i �� Ci

Ori �� C 00i OC 00i�Owi

��

C 0 ��jCi

��

�w0

i

��

CrCi

��

�wi

��

LC 00i

�Lwi

��

C 00�wCi

��

�w00

i

��

such that the induced map ji[A0 jBi W A[A0B 0 � B is a cofibration. Notice that six ofthe eight weak cofiber sequences in this diagram are already in the six diagrams above.The other two weak cofiber sequences are just assumed to exist with the property ofmaking the diagram commutative.

Theorem 3.1. In the situation above the following equation holds

˚�A

� � ˚�B

�C ˚�C

� D ˚�0

� � f�g C ˚�00

�: (S1)

Proof. For i D 1; 2 let �Ai , �Bi , �Ci , �0i , �i , and �00i be the upper and the lower weakcofiber sequences of the pairs above, respectively. By Proposition 1.6

� Œ�A1 � � Œ�C1 �ŒA� � Œ�1�C Œ�B1 �C Œ�001�ŒB0� C Œ�01� D hŒC 0�; ŒA00�i

D �Œ�A2 � � Œ�C2 �ŒA� � Œ�2�C Œ�B2 �C Œ�002�ŒB0� C Œ�02�:

Since @.�C1 / D @.�C2 / and @.�001/ D @.�002/ the actions cancel,

�Œ�A1 �� Œ�C1 �� Œ�1�C Œ�B1 �C Œ�001�C Œ�01� D �Œ�A2 �� Œ�C2 �� Œ�2�C Œ�B2 �C Œ�002�C Œ�02�:

Now one can rearrange the terms in this equation, by using that pairs of weak cofibersequences are central in D1C, obtaining the equation in the statement.

Theorem 3.1 encodes the most relevant relation satisfied by pairs of weak cofibersequences inK1C. They satisfy a further relation which follows from the very definitionin (2.1), but which we would like to record as a proposition by analogy with [6].


Proposition 3.2. A pair of weak cofiber sequences given by two times the same weakcofiber sequence is trivial. 8

<ˆ:

C1 �� w��

A��j��

��j�� B

r ��

r ��

� C

C1 �w ��

9>>=>>;D 0: (S2)

We establish a further useful relation in the following proposition.

Proposition 3.3. Relations (S1) and (S2) imply that the sum of two pairs of weakcofiber sequences coincides with their coproduct.8

ˆ<ˆ:

C1 _ NC1 �� A _ NA ��

�� B _ NB��

��C _ NC

C2 _ NC2!! ��

9>>>=>>>;D

8<ˆ:

C1 �� A

�� B

��

�� C

C2 ��

9>>=>>;

C

8ˆ<ˆ:

NC1 "" ��NA �� NB

��

��

NCNC2

## �

9>>>=>>>;:

Proof. Apply relations (S1) and (S2) to the following pairs of weak cofiber sequences

C1 �� A

�� B

��

C;

C2 ��

C1 _ NC1 �� A _ NA ��

�� B _ NB��

�� C _ NC ;C2 _ NC2

!! ��

NC1 �� NA �� NB

��

��

NC ;NC2

## �

NA ""

�1��

A��i1 ��i1

�� A _ NAp2 $$ $$��

p2��

��NA;

NA##�1

��

NB ""

�1

B��i1 ��i1

�� B _ NBp2 $$ $$��

p2��

��NB;

NB##�1

��

NC ��

�1��

C��i1 ��i1

�� C _ NCp2 %% %%��

p2��

��NC :

NC��1

4 Waldhausen categories with functorial coproducts

Waldhausen categories admit finite coproducts A _ B . The operation _ need not bestrictly associative and unital. However, as we check below, any Waldhausen categorycan be replaced by an equivalent one with strict coproducts.


We now give an explicit definition of what we understand by a strict coproductfunctor. Let C be a Waldhausen category endowed with a symmetric monoidal structureC which is strictly associative

.AC B/C C D AC .B C C/;strictly unital with unit object �, the distinguished zero object,

� C A D A D AC �;but not necessarily strictly commutative, such that

A D AC � �! AC B � �C B D Bis always a coproduct diagram. Such a category will be called a Waldhausen categorywith a functorial coproduct. Then we define the sum-normalized stable quadraticmodule DC� C as the quotient of D�C by the extra relation

(R10) ŒBi2� AC B p1� A� D 0.

Remark 4.1. In DC� C, the relations (R2) and (R10) imply that

ŒAC B� D ŒA�C ŒB�:Furthermore, relation (R9) becomes equivalent to

(R90) hŒA�; ŒB�i D Œ�B;A W B C A Š�! AC B�.Here �B;A is the symmetry isomorphism of the symmetric monoidal structure. Thisequivalence follows from the commutative diagram

A ��i2 �� B C A p1 ��

�B;A Š��

B

A ��i1

�� AC Bp2

�� B

together with (R4), (R7) and (R10).

Theorem 4.2. Let C be a Waldhausen category with a functorial coproduct such thatthere exists a set S which freely generates the monoid of objects of C under the oper-ationC. Then the projection

p W D�C � DC� C

is a weak equivalence. Indeed, it is part of a strong deformation retraction.

We prove this theorem later in this section.Waldhausen categories satisfying the hypothesis of Theorem 4.2 are general enough

as the following proposition shows.


Proposition 4.3. For anyWaldhausen category C there is anotherWaldhausen categorySum.C/ with a functorial coproduct whose monoid of objects is freely generated bythe objects of C except from �. Moreover, there are natural mutually inverse exactequivalences of categories

Sum.C/'

�� C .

��

Proof. The statement already says which are the objects of Sum.C/. The functor 'sends an object Y in Sum.C/, which can be uniquely written as a formal sum of non-zero objects in C, Y D X1 C � � � C Xn, to an arbitrarily chosen coproduct of theseobjects in C

'.Y / D X1 _ � � � _Xn:For n D 0; 1 we make special choices, namely for n D 0, '.Y / D �, and for n D 1

we set '.Y / D X1. Morphisms in Sum.C/ are defined in the unique possible way sothat ' is fully faithful. Then the formal sum defines a functorial coproduct on Sum.C/.The functor sends � to the zero object of the free monoid and any other object in Cto the corresponding object with a single summand in Sum.C/, so that ' D 1. Theinverse natural isomorphism 1 Š ',

X1 C � � � CXn Š .X1 _ � � � _Xn/;is the unique isomorphism in Sum.C/ which ' maps to the identity. Cofibrationsand weak equivalences in Sum.C/ are the morphisms which ' maps to cofibrationsand weak equivalences, respectively. This Waldhausen category structure in Sum.C/makes the functors ' and exact.

Let us recall the notion of homotopy in the category of stable quadratic modules.

Definition 4.4. Given morphisms f; g W C� ! C 0� of stable quadratic modules, a

homotopy f˛' g from f to g is a function ˛ W C0 ! C 01 satisfying

1. ˛.c0 C d0/ D ˛.c0/f0.d0/ C ˛.d0/,2. g0.c0/ D f0.c0/C @˛.c0/,3. g1.c1/ D f1.c1/C ˛@.c1/.The following lemma then follows from the laws of stable quadratic modules.

Lemma 4.5. A homotopy ˛ as in Definition 4.4 satisfies

˛.Œc0; d0�/ D �hf0.d0/; f0.c0/i C hg0.d0/; g0.c0/i; (a)

˛.c0/C g1.c1/ D f1.c1/C ˛.c0 C @.c1//: (b)

Now we are ready to prove Theorem 4.2.


Proof of Theorem 4.2. In order to define a strong deformation retraction,

D�C˛

&&p

��DC� C,

j�� 1

˛' jp; pj D 1;

the crucial step will be to define the homotopy ˛ W D0C! D1C on the generators ŒA�of D0C. Then one can use the equations

1. ˛.c0 C d0/ D ˛.c0/d0 C ˛.d0/,2. jp.c0/ D c0 C @˛.c0/,3. jp.c1/ D c1 C ˛@.c1/,

for ci ; di 2 DiC to define the composite jp and the homotopy ˛ on all of D�C. It isa straightforward calculation to check that the map jp so defined is a homomorphismand that ˛ is well defined. In particular, Lemma 4.5 (a) implies that ˛ vanishes oncommutators of length 3. In order to define j we must show that jp factors throughthe projection p, that is,

jp.ŒBi2� AC B p1� A�/ D 0:

If we also show thatp˛ D 0

then equations (2) and (3) above say pjp.ci / D p.ci /C 0, and since p is surjective itfollows that the composite pj is the identity.

The set of objects of C is the free monoid generated by objects S 2 S. Thereforewe can define inductively the homotopy ˛ by

˛.ŒS C B�/ D ŒB i2� S C B p1� S�C ˛.ŒB�/for B 2 C and S 2 S, with ˛.ŒS�/ D 0. We claim that a similar relation then holds forall objects A, B of C,

˛.ŒAC B�/ D ŒB i2� AC B p1� A�C ˛.ŒA�C ŒB�/ (4.1)

D ŒB i2� AC B p1� A�C ˛.ŒA�/ŒB� C ˛.ŒB�/:If A D � or A 2 S then (4.1) holds by definition, so we assume inductively it holds forgiven A, B and show it holds for S C A, B also:

˛.ŒS C AC B�/ D ŒAC B � S C AC B � S�C ˛.ŒAC B�/D ŒAC B � S C AC B � S�C ŒB � AC B � A�C ˛.ŒA�/ŒB� C ˛.ŒB�/D ŒB � S C AC B � S C A�C ŒA � S C A � S�ŒB� C ˛.ŒA�/ŒB� C ˛.ŒB�/D ŒB � S C AC B � S C A�C ˛.ŒS C A�/ŒB� C ˛.ŒB�/:


Here we have used (R8) for the composable cofibrations

B � AC B � S C AC B:It is clear from the definition of ˛ that the relation p˛ D 0 holds. It remains to see thatjp factors through DC� C, that is,

jp.c1/ D 0 if c1 D ŒB � AC B � A�:

Let c0 D ŒAC B� so that c0 C @.c1/ D ŒA�C ŒB�. Then by Lemma 4.5 (b) we have

jp.c1/ D �˛.c0/C c1 C ˛.c0 C @.c1//D � ˛.ŒAC B�/C ŒB � AC B � A�C ˛.ŒA�C ŒB�/:

This is zero by (4.1).

We finish this section with four lemmas which show useful relations in DC� C.

Lemma 4.6. The following equality holds in DC� C.

ŒAC A0 � B C B 0 � B=AC B 0=A0 � C C C 0�D ŒA � B � B=A

� C �ŒB0� C ŒA0 � B 0 � B 0=A0 � C 0�C hŒA�; ŒC 0�i:

Proof. Use (R4), (R10) and Proposition 1.6 applied to the commutative diagram

A0 ��

��

B 0 ��

��

B 0=A0��

��

C 0��

��

AC A0 ��

��

B C B 0 ��

��

B=AC B 0=A0

��

C C C 0��

��

A �� B �� B=A C��

A �� B �� B=A C .��

As special cases we have

Lemma 4.7. The following equality holds in DC� C.

ŒAC A0 � B C B 0 � B=AC B 0=A0�D ŒA � B � B=A�ŒB

0� C ŒA0 � B 0 � B 0=A0�C hŒA�; ŒB 0=A0�i:Lemma 4.8. The following equality holds in DC� C.

ŒAC B �! A0 C B 0� D ŒA �! A0�ŒB0� C ŒB �! B 0�:


We generalize (R90) in the following lemma.

Lemma 4.9. Let C be a Waldhausen category with a functorial coproduct and letA1; : : : ; An be objects in C. Given a permutation � 2 Sym.n/ of n elements we denoteby

�A1;:::;An W A�1 C � � � C A�nŠ�! A1 C � � � C An

the isomorphism permuting the factors of the coproduct. Then the following formulaholds in DC� C:

Œ�A1;:::;An � DXi>j�i<�j

hŒA�i �; ŒA�j �i:

Proof. The result holds for n D 1 by (R4). Suppose � 2 Sym.n/ for n � 2, and notethat the isomorphism �A1;:::;An factors naturally as

.� 0A1;:::;An�1C 1/.1C �An;B/ W A�1 C � � � C A�n �! A� 0

1C � � � C A� 0

n�1C An

�! A1 C � � � C An�1 C An

where .� 01; : : : ; � 0n�1; n/ D .�1; : : : ; b�k; : : : ; �n; �k/ and B D A�kC1C � � � C A�n .

Therefore by (R4), (R6) and Lemma 4.8,

Œ�A1;:::;An � D Œ� 0A1;:::;An�1C 1�C Œ1C �An;B �

D .Œ� 0A1;:::;An�1�ŒAn� C 0/C .0C Œ�An;B �/:

By induction, (R90) and Remark 4.1 this is equal toXp>q� 0

p<�0

q

hŒA� 0

p�; ŒA� 0

q�i C hŒB�; ŒAn�i D

Xi>j

�i<�j<n

hŒA�i �; ŒA�j �i CXi>j

�i<�jDn

hŒA�i �; ŒA�j �i

as required.

5 Proof of Theorem 2.1

The morphism of stable quadratic modules D�F W D�C! D�D induced by an exactfunctor F W C ! D, see [4], takes pairs of weak cofiber sequences to pairs of weakcofiber sequences,

.D�F /

8<ˆ:

C1 �� A

�� B

��

�� C

C2 ��

9>>=>>;D

8ˆ<ˆ:

F.C1/'' ��

F.A/�� F.B/

(( ((��

)) ))�� F.C/

F.C2/** ��

9>>>=>>>;:

By [4, Theorem 3.2] exact equivalences of Waldhausen categories induce homotopyequivalences of stable quadratic modules, and hence isomorphisms in K1. Therefore


if the theorem holds for a certain Waldhausen category then it holds for all equivalentones. In particular by Proposition 4.3 it is enough to prove the theorem for any Wald-hausen category C with a functorial coproduct such that the monoid of objects is freelygenerated by a set S. By Theorem 4.2 we can work in the sum-normalized constructionDC� C in this case.

Any element x 2 DC1 C is a sum of weak equivalences and cofiber sequences withcoefficients C1 or �1. By Lemma 4.8 modulo the image of h � ; � i we can collect onthe one hand all weak equivalences with coefficient C1 and on the other all weakequivalences with coefficient �1. Moreover, by Lemma 4.7 we can do the same forcofiber sequences. Therefore the following equation holds modulo the image of h � ; � i.

x D �ŒV1 �! V2� � ŒX1 � X2 � X3�

C ŒY1 � Y2 � Y3�C ŒW1 �! W2�

(R4), (R10), 4.8, 4.7 D �ŒV1 CW1 �! V2 CW1�� ŒX1 C Y1 � X2 C Y3 C Y1 � X3 C Y3�C ŒX1 C Y1 � X3 CX1 C Y2 � X3 C Y3�C ŒV1 CW1 �! V1 CW2�

renaming D �ŒL �! L1� � ŒA � E1 � D�

C ŒA � E2 � D�C ŒL �! L2� mod h � ; � i:Suppose that @.x/ D 0 modulo commutators. Then modulo Œ � ; � �

0 D �ŒL�C ŒL1� � ŒA� � ŒD�C ŒE1�� ŒE2�C ŒD�C ŒA� � ŒL2�C ŒL�

D ŒL1�C ŒE1� � ŒE2� � ŒL2� mod Œ � ; � �;i.e. ŒL1 CE1� D ŒL2 CE2� mod Œ � ; � �:

(5.1)

The quotient of DC0 C by the commutator subgroup is the free abelian group on S,hence by (5.1) there are objects S1; : : : ; Sn 2 S and a permutation � 2 Sym.n/ suchthat

L1 CE1 D S1 C � � � C Sn;L2 CE2 D S�1 C � � � C S�n ;

so there is an isomorphism

�S1;:::;Sn W L2 CE2��! L1 CE1:

Again by Lemmas 4.8 and 4.7 modulo the image of h � ; � ix D �ŒLCD �! L1 CD� � ŒA � L1 CE1 � L1 CD�

C ŒA � L2 CE2 � L2 CD�C ŒLCD �! L2 CD�4.9 D �ŒLCD �! L1 CD� � ŒA � L1 CE1 � L1 CD�C


C Œ�S1;:::;Sn W L2 CE2��! L1 CE1�

C ŒA � L2 CE2 � L2 CD�C ŒLCD �! L2 CD�(R7) D �ŒLCD �! L1 CD� � ŒA � L1 CE1 � L1 CD�

C ŒA � L1 CE1 � L2 CD�C ŒLCD �! L2 CD�renaming D �ŒC �! C1� � ŒA � B � C1�

C ŒA � B � C2�C ŒC �! C2�

D �ŒC �! C1�ŒA� � ŒA � B � C1�

C ŒA � B � C2�C ŒC �! C2�ŒA�

D �ŒA � B � C1� C �C ŒA � B � C2

� C � mod h � ; � i;i.e. there is y 2 DC1 C in the image of h � ; � i such that

x D

8<ˆ:

C1 �� A

�� B

��

�� C

C2 ��

9>>=>>;C y:

Now assume that @.x/ D 0. Since the pair of weak cofiber sequences is also inthe kernel of @ we have @.y/ D 0. In order to give the next step we need a technicallemma.

Lemma 5.1. Let C� be a stable quadratic module such that C0 is a free group ofnilpotency class 2. Then any element y 2 Ker @\ Imageh � ; � i is of the form y D ha; aifor some a 2 C0.

Proof. For any abelian group A let y 2A be the quotient of the tensor square A˝A bythe relations a ˝ b C b ˝ a D 0, a; b 2 A, and let ^2A be the quotient of A˝ A bythe relations a ˝ a D 0, a 2 A, which is also a quotient of y 2A. The projection ofa˝ b 2 A˝ A to y 2A and ^2A is denoted by a y b and a ^ b, respectively.

There is a commutative diagram of group homomorphisms

C ab0 ˝ C ab

0

�� ++ ++

��

h � ;� i��

��

C ab0 ˝ Z=2 � � N� �� y 2C ab

0

q��

c1

��

^2C ab0

c0

��

C1@ �� C0

where N�.a˝ 1/ D a y a, the factorization c1 of h � ; � i is given by c1.a y b/ D ha; bi,which is well defined by Definition 1.1 (3), and c0.a^ b/ D Œb; a� is well known to beinjective in the case C0 is free of nilpotency class 2.


Moreover, C ab0 is a free abelian group, hence the middle row is a short exact se-

quence, see [1, I.4]. Therefore any element y 2 C1 which is both in the image of c1and the kernel of @ is in the image of c1 N� as required.

The group DC0 C is free of nilpotency class 2, therefore by the previous lemmay D ha; ai for some a 2 DC0 C. By the laws of a stable quadratic module the elementha; ai only depends on a mod 2, compare [4, Definition 1.8], and so we can supposethat a is a sum of basis elements with coefficient C1. Hence by Remark 4.1 we cantake a D ŒM � for some object M in C,

y D hŒM �; ŒM �i:The element hŒM �; ŒM �i is itself a pair of weak cofiber sequences

hŒM �; ŒM �i D

8<ˆ:

M �� 1��

M��i2 ��i1

�� M CMp1 ��

p2 !!

!!M

M �1 ��

9>>=>>;;

therefore x is also a pair of weak cofiber sequences, by Proposition 3.3, and the proofof Theorem 2.1 is complete.

6 Comparison with Nenashev’s approach for exact categories

For any Waldhausen category C we denote by Kwcs1 C the abelian group generated by

pairs of weak cofiber sequences8<ˆ:

C1 �� A

�� B

��

�� C

C2 ��

9>>=>>;

modulo the relations (S1) and (S2) in Section 3. Theorem 3.1 and Proposition 3.2 showthe existence of a natural homomorphism

Kwcs1 C � K1C (6.1)

which is surjective by Theorem 2.1.Given an exact category E Nenashev defines in [6] an abelian group D.E/ by

generators and relations which surjects naturally to K1E. Moreover, he shows in [7]that this natural surjection is indeed a natural isomorphism

D.E/ Š K1E: (6.2)

Generators of D.E/ are pairs of short exact sequencesÁ

��j1 ��j2

�� Br1��

r2�� C

μ:


They satisfy two kind of relations. The first, analogous to (S2), says that a generatorvanishes provided j1 D j2 and r1 D r2. The second is a simplification of (S1): givensix pairs of short exact sequences

A0 ��jA1 ��

jA2

�� A

rA1��

rA2

�� A00; B 0 ��jB1 ��

jB2

�� B

rB1��

rB2

�� B 00; C 0 ��jC1 ��

jC2

�� C

rC1��

rC2

�� C 00;

A0 ��j 0

1 ��

j 0

2

�� B 0r 0

1��

r 0

2

�� C 0; A��j1 ��j2

�� Br1��

r2�� C; A00 ��

j 00

1 ��

j 00

2

�� B 00r 00

1��

r 00

2

�� C 00;

denoted for simplicity as �A, �B , �C , �0, �, and �00, such that the diagram

A0 ��jAi ��

��

j 0

i

��

A��

ji

��

rAi �� A00��

j 00

i

��

B 0 ��jBi ��

r 0

i��

BrBi ��

ri��

B 0ir 00

i��

C 0 ��jCi

�� CrCi

�� C 00

commutes for i D 1; 2 then˚�A

� � ˚�B

�C ˚�C

� D ˚�0

� � f�g C ˚�00

�:

Proposition 6.1. For any exact category E there is a natural isomorphism

D.E/ Š Kwcs1 E:

Proof. There is a clear homomorphism D.E/! Kwcs1 E defined by

Á

��j1 ��j2

�� Br1��

r2�� C

μ7!

8<ˆ:

C �� 1��

A��j1 ��j2

�� B

r1 ��

r2

C

C��1

��

9>>=>>;:

Exact categories regarded as Waldhausen categories have the particular feature thatall weak equivalences are isomorphisms, hence one can also define a homomorphismKwcs1 E! D.E/ as

8<ˆ:

C1 �� Šw1

��

A��j1 ��j2

�� B

r1 ��

r2 ��

� C

C2

w2

Š��

9>>=>>;7!

Á

��j1 ��j2

�� B

w�11r1��

w�12r2

�� C

μ:

We leave the reader to check the compatibility with the relations and the fact that thesetwo homomorphisms are inverse of each other.


Remark 6.2. The composite of the isomorphism in Proposition 6.1 with the naturalepimorphism (6.1) for C D E coincides with Nenashev’s isomorphism (6.2), thereforethe natural epimorphism (6.1) is an isomorphism when C D E is an exact category.

7 Weak cofiber sequences and the stable Hopf map

After the previous section one could conjecture that the natural epimorphism

Kwcs1 C � K1C

in (6.1) is an isomorphism not only for exact categories but for any Waldhausen categoryC. In order to support this conjecture we show the following result.

Theorem 7.1. For any Waldhausen category C there is a natural homomorphism

W K0C˝ Z=2 �! Kwcs1 C

which composed with (6.1) yields the homomorphism � � W K0C˝Z=2! K1C deter-mined by the action of the stable Hopf map � 2 �s� in the stable homotopy groups ofspheres.

For the proof we use the following lemma.

Lemma 7.2. The following relation holds in Kwcs1 C

8<ˆ:

A00 ��w0

1w1

��

� �� A00

1 ,, ,,"""

1�� ##

# A

A00��w0

2w2

��

9>>=>>;D

8<ˆ:

A0 ��w1

� �� A0

1 ,, ,,��

1��

� A

A0��w2

��

9>>=>>;

C

8<ˆ:

A00 ��w0

1��

� �� A00

1 ,, ,,"""

1�� ##

# A0

A00�w0

2

��

9>>=>>;:

Proof. Use relations (S1) and (S2) applied to the following pairs of weak cofiber se-quences

� ��

��

� ��

��

�;� ��

A00 ��w0

1w1

��

� �� A00

1 ,, ,,��

1��

� A;

A00�w0

2w2

��

A0 ��w1��

� �� A0

1 ��

1 ��

� A;

A0��w2��

� ��

��

� ��

��

�;� ��

A00 ��w0

1��

� �� A00

1 ,, ,,��

1��

� A0;

A00�w0

2

��

A ��

�1��

� �� A

1 ��

1 �

�� A:

A��1��


Now we are ready to prove Theorem 7.1.

Proof of Theorem 7.1. We define the homomorphism by

ŒA� D

8<ˆ:

A ��

�1��

A��i2 ��i1

�� A _ Ap1 %% %%$$$$

p2�� %%

%% A

A��1

9>>=>>;D

8<ˆ:

A _ A ��

�1&&&

� �� A _ A

1 �� '''

1 &&&

A _ AA _ A

!!��A;A'''

9>>=>>;:

Here the second equality follows from (S1) and (S2) applied to the following pairs ofweak cofiber sequences

A ��

�1

� �� A

1 ��

1��

A;

A��1��

A _ A ��

�1��

� �� A _ A

1 ��

1 ��

� A _ A;A _ A

!!��A;A��

A ��

�1

� �� A

1 ��

1��

A;

A��1��

� ��

��

� ��

��

�;� ��

A ��

�1

A��i2 ��i1

�� A _ Ap1 %% %%��

p2 ��

A;

A��1��

A ��

�1��

A��i2 ��i2

�� A _ Ap1 %% %%$$$$

p1�� %%

%% A:

A��1��

Given a cofiber sequenceAj

� Br� C the equation ŒC �CŒA� D ŒB� follows

from (S1) and (S2) applied to the following pairs of weak cofiber sequences

A ��

�1

A��i2 ��i1

�� A _ Ap1 %% %%��

p2 ��

A;

A��1��

B ��

�1��

B��i2 ��i1

�� B _ Bp1 %% %%��

p2 �� %%%%

B;

B��1��

C ��

�1��

C��i2 ��i1

�� C _ Cp1 %% %%$$$$

p2 �� %%%%

C;

C��1��

C ��

�1��

A��j��

��j�� B

r ��

r �

�� C;

C��1��

C _ C ��

�1&&&

A _ A ��j_j

��j_j

�� B _ Br_r �� '''

r_r &&&

C _ C;C _ C

!!�1'''

C ��

�1��

A��j��

��j�� B

r ��

r

C:

C�1��

Given a weak equivalence w W A ! A0 the equation ŒA� D ŒA0� follows from(S1) and (S2) applied to the following pairs of weak cofiber sequences

� ��

��

� ��

��

�;� ��

A0 ��1��

A0 ��i2 ��i1

�� A0 _ A0p1 (( (($$$$

p2 )) ))%%%%

A0;

A0�1��

A ��

�1

A��i2 ��i1

�� A _ Ap1 %% %%��

p2 ��

A;

A��1��

A0 ��w��

� �� A0

1 ��

1 ��

� A;

A0��w��

A0 _ A0 ��w_w&&&

� �� A0 _ A0

1 -- --'''

1�� &&

& A _ A;A0 _ A0

!!�w_w'''

A0 ��w��

� �� A0

1 ,, ,,��

1��

� A:

A0�w��


The equation 2ŒA� D 0 follows from (S2) and Lemma 7.2 by using the secondformula for ŒA� above and the fact that �2A;A D 1.

We have already shown that is a well-defined homomorphism. The composite of and (6.1) coincides with the action of the stable Hopf map by the main result of [4].

Appendix. Free stable quadratic modules and presentations

Free stable quadratic modules, and also stable quadratic modules defined by generatorsand relations, can be characterized up to isomorphism by obvious universal properties.In this appendix we give more explicit constructions of these notions.

Let squad be the category of stable quadratic modules and let

U W squad �! Set � Set

be the forgetful functor, U.C�/ D .C0; C1/. The functor U has a left adjoint F , and astable quadratic module F.E0; E1/ is called a free stable quadratic module on the setsE0 andE1. In order to give an explicit description of F.E0; E1/we fix some notation.Given a set E we denote by hEi the free group with basis E, and by hEiab the freeabelian group with basisE. The free group of nilpotency class 2with basisE, denotedby hEinil, is the quotient of hEi by triple commutators. Given a pair of sets E0 andE1, we write E0 [ @E1 for the set whose elements are the symbols e0 and @e1 for eache0 2 E0, e1 2 E1.

To define F.E0; E1/, consider the groups

F.E0; E1/0 D hE0 [ @E1inil;

F .E0; E1/1 D .hE0iab ˝ Z=2/ � Ker ı:

Here ı W F.E0; E1/0 ! hE0iab is the homomorphism given by ıe0 D e0 and ı@e1 D 0.In the notation of the proof of lemma 5.1 there are isomorphisms

Ker ı Š ^2hE0iab � hE0 �E1iab � hE1inil;

.hE0iab ˝ Z=2/ � Ker ı Š O 2hE0iab � hE0 �E1iab � hE1inil;

and intuitively we think of F.E0; E1/1 as a group generated by symbols he0; e00i,he0; @e1i and e1. The symbol h@e1; @e01i is unnecessary since it will be given by thecommutator Œe01; e1�.

We define structure homomorphisms on F.E0; E1/ as follows. The boundary

@ W F.E0; E1/1 �! F.E0; E1/0

is the projection onto Ker ı followed by the inclusion of the kernel. The bracket

h � ; � i W F.E0; E1/ab0 ˝ F.E0; E1/ab

0 �! F.E0; E1/1


is given by the product of the following two homomorphisms,

c0 W F.E0; E1/ab0 ˝ F.E0; E1/ab

0 �! hE0iab ˝ Z=2;

defined on the generators x; y 2 E0 [ @E1 by c0.x; y/ D x ˝ 1 if x D y 2 E0 andc0.x; y/ D 0 otherwise, and

c W F.E0; E1/ab0 ˝ F.E0; E1/ab

0 �! Ker ı

induced by the commutator bracket, c.a; b/ D Œb; a�.It is now straightforward to define explicitly the stable quadratic module C� pre-

sented by generators Ei and relations Ri � F.E0; E1/i in degrees i D 0; 1, by

C0 D F.E0; E1/0=N0;C1 D F.E0; E1/1=N1:

Here N0 � F.E0; E1/0 is the normal subgroup generated by the elements of R0 and@R1, and N1 � F.E0; E1/1 is the normal subgroup generated by the elements ofR1 and hF.E0; E1/0; N0i. The boundary and bracket on F.E0; E1/ induce a stablequadratic module structure on C� which satisfies the following universal property:given a stable quadratic module C 0�, any pair of functions Ei ! C 0i .i D 0; 1/ suchthat the induced morphism F.E0; E1/! C 0� annihilates R0 and R1 induces a uniquemorphism C� ! C 0�.

In [1] Baues considers the totally free stable quadratic module C� with basis givenby a function g W E1 ! hE0inil. In the language of this paper C� is the stable quadraticmodule with generatorsEi in degree i D 0; 1 and degree 0 relations @.e1/ D g.e1/ forall e1 2 E1.

References

[1] H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, de Gruyter Exp.Math. 2, Walter de Gruyter, Berlin 1991.

[2] H. Gillet, D. R. Grayson, The loop space of theQ-construction, Illinois J. Math. 31 (1987),574–597.

[3] T. Gunnarsson, R. Schwänzl, R. M. Vogt, F. Waldhausen, An un-delooped version of alge-braic K-theory, J. Pure Appl. Algebra 79 (1992), 255–270.

[4] F. Muro, A. Tonks, The 1-type of a WaldhausenK-theory spectrum, Adv. Math. 216 (2007),178–211, .

[5] A. Nenashev, Double short exact sequences produce all elements of Quillen’sK1, in Alge-braic K-theory (Poznan, 1995), Contemp. Math. 199, Amer. Math. Soc., Providence, R.I.,1996, 151–160.

[6] A. Nenashev, Double short exact sequences and K1 of an exact category, K-Theory 14(1998), 23–41.

[7] A. Nenashev, K1 by generators and relations, J. Pure Appl. Algebra 131 (1998), 195–212.


[8] C. Sherman, On K1 of an abelian category, J. Algebra 163 (1994), 568–582.

[9] C. Sherman, On K1 of an exact category, K-Theory 14 (1998), 1–22.

[10] F. Waldhausen, Algebraic K-theory of topological spaces. I, in Algebraic and geometrictopology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, Proc.Sympos. Pure Math. 23, Amer. Math. Soc., Providence, R.I., 1978, 35–60.

[11] F. Waldhausen, Algebraic K-theory of spaces, In Algebraic and geometric topology (NewBrunswick, N.J., 1983), Lecture Notes in Math. 1126, Springer-Verlag, Berlin 1985,318–419.

[12] C. Weibel, An introduction to algebraicK-theory, book in progress available at http://math.rutgers.edu/�weibel/Kbook.html.

Twisted K -theory – old and new

by Max Karoubi

Some history and motivation about this paper

The subject “K-theory with local coefficients”, now called “twisted K-theory”, wasintroduced by P. Donovan and the author in [19] almost forty years ago.1 It associatesto a compact space X and a “local coefficient system”

˛ 2 GBr.X/ D Z=2 �H 1.X IZ=2/ � Tors.H 3.X IZ//an abelian groupK˛.X/which generalizes the usual Grothendieck–Atiyah–HirzebruchK-theory of X when we restrict ˛ being in Z=2 (cf. [5]). This “graded Brauer group”GBr.X/ has the following group structure: if ˛ D ."; w1; W3/ and ˛0 D ."0; w01; W 03/are two elements, one defines the sum˛C˛0 as ."C"0; w1Cw01; W3CW 03Cˇ.w1 �w01//,where ˇ W H 2.X IZ=2/ ! H 3.X IZ/ is the Bockstein homomorphism. With thisdefinition, one has a generalized cup-product2:

K˛.X/ �K˛0

.X/! K˛C˛0

.X/:

The motivation for this definition is to give in K-theory a satisfactory Thom isomor-phism and Poincaré duality pairing which are analogous to the usual ones in cohomologywith local coefficients. More precisely, as proved in [28], if V is a real vector bundleon a compact space X with a positive metric, the K-theory of the Thom space of V isisomorphic to a certain “algebraic” group KC.V /.X/ associated to the Clifford bundleC.V /, viewed as a bundle of Z=2�graded algebras. A careful analysis of this groupshows that it depends only on the class of C.V / in GBr.X/, the three invariants beingrespectively the rank of V mod. 2, w1.V / and ˇ.w2.V // D W3.V /, where w1.V /,w2.V / are the first two Stiefel–Whitney classes of V . In particular, if V is a cspinorialbundle of even rank, one recovers a well-known theorem of Atiyah and Hirzebruch.On the other hand, if X is a compact manifold, it is well known that such a Thomisomorphism theorem induces a pairing between K-groups

K˛.X/ �K˛0

.X/! Z

if ˛ C ˛0 is the class of �C.V / in GBr.X/, where V is the tangent bundle of X .The necessity to revisit these ideas comes from a new interest in the subject because

of its relations with Physics [43], as shown by the number of recent publications.However, for these applications, the first definition recalled above is not completesince the coefficient system is restricted to the torsion elements of H 3.X IZ/. Asit was pointed out by J. Rosenberg [38] and later on by C. Laurent, J.-L. Tu, P. Xu

1See the appendix for a short history of the subject.2Strictly speaking, this product is defined up to non canonical isomorphism ; see 2.1 for more details.

118 M. Karoubi

[35], M. F. Atiyah and G. Segal [8], this restriction is in fact not necessary. In orderto avoid it, one may use for instance the Atiyah–Jänich theorem [2], [27] about therepresentability of K-theory by the space of Fredholm operators (already quoted in[19] for the cup-product which cannot be defined otherwise).

In the present paper, we would like to make a synthesis between different viewpointson the subject: [19], [38], [35] and [8] (partially of course)3. We hope to have been“pedagogical” in some sense to the non experts.

However, this paper is not just historical. It presents the theory with another pointof view and contains some new results. We extend the Thom isomorphism to this moregeneral setting (see also [16]), which is important in order to relate the “ungraded”and “graded” twisted K-theories. We compute many interesting equivariant twistedK-groups, complementing the basic papers [35], [8] and [9]. For this purpose, we usethe “Chern character” for finite group actions, as defined by Baum, Connes, Kuhn andSłominska [11], [33], [41], together with our generalized Thom isomorphism. Theselast computations are related to some previous ones [31] and to the work of manyauthors. Finally, we introduce new cohomology operations which are complementaryto those defined in [19] and [9].

We don’t pretend to be exhaustive in a subject which has already many ramifications.In an appendix to this paper we try to give a short historical survey and a list of interestingcontributions of many authors related to the results quoted here.

General plan of the paper

Let us first recall the point of view developed in [19], in order to describe the backgroundmaterial. We consider a locally trivial bundle of Z=2-graded central simple complexalgebrasA, i.e. modelled onM2n.C/ orMn.C/�Mn.C/, with the obvious gradings4.Then A has a well-defined class ˛ in the group GBr.X/ (as introduced above). Onthe other hand, one may consider the category of “A-bundles” , whose objects arevector bundles provided with an A-module structure (fibrewise). We call this categoryEA.X/; the graded objects of this category are vector bundles which are modules overA y C 0;1, where C 0;1 is the Clifford algebra C � C D CŒx�=.x2 � 1/. The groupK˛.X/ is now defined as the “Grothendieck group” of the forgetful functor

EA yC0;1.X/! EA.X/:

We refer the reader to [28], p. 191, for this definition which generalizes the usualGrothendieck group of a category. For our purpose, we make it quite explicit at the endof §1, using the concept of “grading”.

Despite its algebraic simplicity, this definition of K˛.X/ is not quite satisfactoryfor various reasons. For instance, it is not clear how to define in a simple way a

3As it was pointed out to me by J. Rosenberg, one should also add the following reference, in the spirit of[18]: Ellen Maycock Parker, The Brauer group of graded continuous trace C*-algebras, Trans. Amer. Math.Soc. 308 (1988), Nr. 1, 115–132.

4up to graded Morita equivalence.

Twisted K-theory – old and new 119

“cup-product”K˛.X/ �K˛0

.X/! K˛C˛0

.X/

as mentioned earlier (even if ˛ and ˛0 are in the much smaller group Z=2).To correct this defect, a second definition may be given in terms of Fredholm

operators in a Hilbert space. More precisely, we consider graded Hilbert bundles Ewhich are also graded A-modules in an obvious sense, together with a continuousfamily of Fredholm operators

D W E ! E

with the following properties:

1. D is self-adjoint of degree 1,

2. D commutes with the action of A (in the graded sense).

One gets an abelian semi-group from the homotopy classes of such pairs .E;D/, withthe addition rule

.E;D/C .E 0;D0/ D .E ˚E 0;D ˚D0/:The associated group gives the second definition of K˛.X/ which is equivalent to thefirst one (see [19], p. 18, and [29], p. 88). We use also the notation KA.X/ instead ofK˛.X/ where ˛ is the class of A in GBr.X/ when we want to be more explicit.

With this new viewpoint, the cup-product alluded to above becomes obvious. It isdefined by the following formula5:

.E;D/Y .E 0;D0/ D .E y E 0;D y 1C 1 y D0/where the symbol y denotes the graded tensor product of bundles or morphisms. It is amap fromKA.X/�KA0

.X/ toKA yA0

.X/ and therefore it induces a (non canonical)map from K˛.X/ �K˛0

.X/ to K˛C˛0

.X/.For simplicity’s sake, we have only considered complex K-theory. We could as

well study the real case: one has to replace GBr.X/ by

GBrO.X/ D Z=8 �H 1.X IZ=2/ �H 2.X IZ=2/:If we take for the coefficient system ˛ D n to be in Z=8, we get the usual groupsKOn.X/ as defined using Clifford algebras in [28] and [29], p. 88 (these groups beingwritten xKn in the later reference).

In this paper, we essentially follow the same pattern, but with bundles of infinitedimension in the spirit of [38]. As a matter of fact, all the technical tools are alreadypresent in [19], [29] and [38], for instance the Fredholm operator machinery which isnecessary to define the cup product. However, this paper is not a rewriting of thesepapers, since we take a more synthetic view point and have other applications in mind.For instance, the K-theory of Banach algebras Kn.A/ and its graded version, denoted

5See the appendix about the origin of such a formula.

120 M. Karoubi

here by GrKn.A/, are more systematically used. On the other hand, since the equiv-ariant K-theory and its relation with cohomology have been studied carefully in [17],[35] and [9], we limit ourselves to the applications in this case. One of them is thedefinition of operations in twisted K-theory in the graded and ungraded situations.

Here is the contents of the paper:

1. K-theory of Z=2-graded Banach algebras

2. Ungraded twisted K-theory in the finite and infinite-dimensional cases

3. Graded twisted K-theory in the finite and infinite-dimensional cases

4. The Thom isomorphism

5. General equivariant K-theory

6. Some computations in the equivariant case

7. Operations in twisted K-theory

Appendix. A short historical survey of twisted K-theory

Acknowledgments. I would like to thank A. Adem, A. Carey, P. Hu, I. Kriz, C. Leruste,V. Mathai, J. Rosenberg, J.-L. Tu, P. Xu and the referee for their remarks and suggestionsafter various drafts of this paper.

1 K -theory of Z=2-graded Banach algebras

HigherK-theory of real or complex Banach algebras A is well known (cf. [28] or [12]for instance). Starting from the usual Grothendieck group K.A/ D K0.A/, there aremany equivalent ways to define “derived functors” Kn.A/, for n 2 Z, such that anyexact sequence of Banach algebras

0 �� A0 �� A �� A00 �� 0

induces an exact sequence of abelian groups

: : : �� KnC1.A/ �� KnC1.A"/ �� Kn.A0/ �� Kn.A/ �� Kn.A"/ �� : : : :

Moreover, by Bott periodicity, these groups are periodic of period 2 in the complexcase and 8 in the real case.

The K-theory of Z=2-graded Banach algebras A (in the real or complex case) isless well known6 and for the purpose of this paper we shall recall its definition whichis already present but not systematically used in [28] and [19]. We first introduce Cp;q

as the Clifford algebra of RpCq with the quadratic form

�.x1/2 � � � � � .xp/2 C .xpC1/2 C � � � C .xpCq/2:6This is of course included in the general KK-theory of Kasparov which was introduced later than our

basic references, at least for C*-algebras.


It is naturally Z=2-graded. IfA is an arbitrary Z=2-graded Banach algebra, Ap;q is thegraded tensor product A y Cp;q . For A unital, we now define the graded K-theory ofA (denoted by GrK.A/) as the K-theory of the forgetful functor:

� W P .A0;1/! P .A/

(see [28] or 1.4 for a concrete definition). Here P .A/ denotes in general the cate-gory of finitely generated projective left A-modules. One may remark that the objectsof P .A0;1/ are graded objects of the category P .A/. The functor � simply “for-gets” the grading. One should also notice that GrK.Ap;q/ is naturally isomorphic toGrK.ApC1;qC1/, since Ap;q is Morita equivalent to ApC1;qC1 (in the graded sense).

If A is not unital, we define GrK.A/ by the usual method, as the kernel of theaugmentation map

GrK.AC/! GrK.k/

where AC is the k-algebra A with a unit added (k D R or C, according to the theory,with the trivial grading).

There is a “suspension functor” on the category of graded algebras, associating toA the graded tensor product A0;1 D A y C 0;1. One of the fundamental results7 in[28], p. 210, is the fact that this suspension functor is compatible with the usual one:in other words, we have a well-defined isomorphism

GrK.A0;1/ � GrK.A.R//

where A.T / denotes in general the algebra of continuous maps f .t/ on the locallycompact space T with values in A, such that f .t/ ! 0 when t goes to infinity. As aconsequence, we have an isomorphism between the following groups (for n � 0):

GrK.A0;n/ � GrK.A.Rn//

which we call GrKn.A/. More generally, we put GrKn.A/ D GrK.Ap;q/ for q�p Dn 2 Z. These groups GrKn.A/ satisfy the same exactness property as the groupsKn.A/ above, from which they are naturally derived. They are of course linked withthem by the following exact sequence (for all n 2 Z):

KnC1.A y C 0;1/ �� KnC1.A/ �� GrKn.A/ �� Kn.A y C 0;1/ �� Kn.A/:

In particular, if we start with an ungraded Banach algebraA, we see that Bott periodicityfollows from these previous considerations, thanks to the periodicity of Clifford algebrasup to graded Morita equivalence: this was the main theme developed in [6], [44] and[28], in order to give a more conceptual proof of the periodicity theorems.

When A is unital, it is technically important to describe the group GrK.A/ in amore concrete way. IfE is an object of P .A/, a grading ofE is given by an involution" which commutes (resp. anticommutes) with the action of the elements of degree 0

7Strictly speaking, one has to replace the category Cp;q with an arbitrary graded category. However, theproof of Theorem 2.2.2 in [28] easily extends to this case.

122 M. Karoubi

(resp. 1) in A. In this way, E with a grading " may be viewed as a module over thealgebraA y C 0;1. We now consider triples .E; "1; "2/, where "1 and "2 are two gradingsof E. The homotopy classes of such triples obviously form a semi-group. Its quotientby the semi-group of “elementary” triples (i.e. such that "1 D "2) is isomorphic toGrK.A/.

2 Ungraded twisted K -theory in the finite andinfinite-dimensional cases

LetX be a compact space and let us consider bundles of algebras A with fiberMn.C/.As it was shown by Serre [25], such bundles are classified by Cech cocycles (up to Cechcoboundaries):

gj i W Ui \ Uj ! PU.n/

where PU.n/ is U.n/=S1, the projective unitary group. In other words, the bundleof algebras A may be obtained by gluing together the bundles of C*-algebras .Ui �Mn.C//, using the transition functions gj i [Note that U.n/ acts on Mn.C/ by innerautomorphisms and therefore induces an action of PU.n/ on the algebraMn.C/]. TheBrauer group of X denoted by Br.X/ is the quotient of this semi-group of bundles(via the tensor product) by the following equivalence relation: A is equivalent to A0iff there exist vector bundles V and V 0 such that the bundles of algebras A y End.V /and A0 y End.V 0/ are isomorphic. It was proved by Serre [25] that Br.X/ is naturallyisomorphic to the torsion subgroup of H 3.X IZ/.

The Serre–Swan theorem (cf. [28] for instance) may be easily translated in thissituation to show that the category of finitely generated projective A-module bundlesE(as in [19]) is equivalent to the category P .A/ of finitely generated projective modulesover A D �.X;A/, the algebra of continuous sections of the bundle A. The keyobservation for the proof is that E is a direct factor of a “trivial” A-bundle; this iseasily seen with finite partitions of unity, sinceX is compact. One should notice that ifA is equivalent to A0 the associated categories P .A/ and P .A0/ are equivalent. Notehowever that this equivalence is non canonical since A˝ End.V / and A0 ˝ End.V 0/are not canonically isomorphic.

Definition 2.1. The ungraded8 twistedK-theoryK.A/.X/ is by definition theK-theoryof the ring A (which is the same as theK-theory of the category EA.X/ mentioned inthe introduction). By abuse of notation, we shall simply call it K.A/. We also defineK.A/n .X/ as theKn-group of the Banach algebra �.X;A/. It only depends on the class

of A in Br.X/ D Tors.H 3.X IZ//.The key observation made by J. Rosenberg [38] is the following: we can “stabilize”

the situation (in the C*-algebra sense), i.e. embed Mn.C/ into the algebra of compactoperators K in a separable Hilbert spaceH , thanks to the split inclusion of Cn in l2.N/.

8We use the notation K.A/.X/, not to be confused with the graded twisted K-group KA.X/ whichwill be defined in the next section.


Now, a bigger group PU.H/ D U.H/=S1 is acting on K by inner automorphisms. Ifwe take a Cech cocycle

gj i W Ui \ Uj ! PU.H/

we may use it to construct a bundle A of (non unital) C*-algebras with fiber K .Let us now consider the commutative diagram

S1 ��

��

S1

��

U.n/ ��

��

U.H/

��

PU.n/ �� PU.H/.

Thanks to Kuiper’s theorem [34], we remark that the classifying space of U.H/ is con-tractible. Therefore, the classifying space BPU.H/ of the topological group PU.H/, isa nice model of the Eilenberg–Mac Lane spaceK.Z; 3/ (compare with the well-knownpaper of Dixmier and Douady [18]). Moreover, if we start with a finite-dimensionalalgebra bundle A over X with fiber Mn.C/, the diagram above shows how to asso-ciate to A another bundle of algebras A0 with fiber K , together with a C*-inclusionfrom A to A0. We note that the invariant W3.A/ in Br.X/ D Tors.H 3.X IZ// definedin [25] is simply induced by the classifying map from X to BPU.H/ (which factorsthrough BPU.n/). In this finite example, it is an n-torsion class since one has anothercommutative diagram

�n

��

�� S1

��

SU.n/

��

�� U.n/

��

PU.n/ PU.n/.

Theorem 2.2. The inclusion from A to A0 induces an isomorphism

Kr.A/ D Kr.�.X;A//! Kr.�.X;A0// D Kr.A0/

where the Kr define the classical topological K-theory of C*-algebras.

Proof. The proof is classical for a trivial algebra bundle, since C is Morita equivalentto K (in the C*-algebra sense). It extends to the general case by a no less classicalMayer–Vietoris argument.

Definition 2.3. Let now A be an algebra bundle with fiber K on a compact space Xwith structural group PU.H/. We define K.A/.X/ (also denoted by K.A/) as theK-theory of the (non unital) Banach algebra �.X;A/. ThisK-theory only depends of

124 M. Karoubi

the class ˛ of A in H 3.X IZ/ and we shall also call it K.˛/.X/ [Due to 2.4, this is ageneralization of definition 2.2].

Before treating the graded case in the next section, we would like to give an equiv-alent definition of K.A/ in terms of Fredholm operators, as was done in [19] for thetorsion elements in H 3.X IZ/ and in [8] for the general case. The basic idea is to re-mark that PU.H/ acts not only on the C*-algebra of compact operators inH , but alsoon the ring of bounded operators End.H/ and on the Calkin algebra End.H/=K (withthe norm topology9)). Let us call B the algebra bundle with fiber End.H/ associatedto the cocycle defined in 2.1 and B=A the quotient algebra bundle. Therefore, we havean exact sequence of C*-algebras bundles

0 �� A �� B �� B=A �� 0.

which induces an exact sequence for the associated rings of sections (thanks to a partitionof unity again)

0 �� .X;A/ �� .X;B/ �� .X;B=A// �� 0.

If B is trivial, it is well known that the algebra of continuous maps from X to End.H/has trivialKn-groups because this algebra is flabby10. By a Mayer–Vietoris argument,it follows that Kn.�.X;B// is also trivial. Therefore the connecting homomorphism

K1.B=A/ D K1.�.X;B=A//! K0.�.X;A// � K.A/.X/ D K.A/is an isomorphism, a well-known observation in index theory.

Let us now consider the elements of B which map onto .B=A/� via the map � .These elements form a bundle of Fredholm operators on H (the twist comes from theaction of PU.H/). This subbundle of B will be denoted by Fredh.H/. Therefore, wehave a principal fibration

�.X;A/ �� .X;Fredh.H// � �� .X; .B=A/�/

with contractible fiber the Banach space �.X;A/ (this fibration admits a local sectionthanks to Michael’s theorem [37]). Therefore, the space of sections of Fredh.H/has the homotopy type of �.X; .B=A/�/. In particular, the path components are inbijective correspondence via the map � . The following theorem is a generalization ofa well-known theorem of Atiyah and Jänich [2], [27]:

Theorem 2.4 ([8]). The set of homotopy classes of continuous sections of the fibration

Fredh.H/ �� X

is naturally isomorphic to K.A/.X/.9other topologies may be also considered, see [8].

10A unital Banach algebra ƒ is called flabby if there exists a continuous functor � from P .ƒ/ to itselfsuch that � C Id is isomorphic to � . For instance, ƒ D End.H/ is flabby since P .A/ is equivalent tothe category of Hilbert spaces which are isomorphic to direct factors inH ; � is then defined by the infiniteHilbert sum �.E/ D E ˚ � � � ˚E ˚ � � � :


Proof. As we have seen above, the two spaces �.X;Fredh.H// and �.X; .B=A/�/have the same homotopy type. On the other hand, it is a well known consequence ofKuiper’s theorem [34] than the (non unital) ring map �.X;B=A/! �.X;Mr.B=A//

induces a bijection between the path components of the associated groups of invert-ible elements (see for instance [32], p. 93). Therefore, �0.�.X;Fredh.H/// may beidentified with

lim�!

r

�0.�.X;GLr.B=A/// D K1.B=A/

and therefore with K.A/, as we already mentioned in 2.6.

Remark 2.5. We may also consider the following “stabilized” bundle

Fredhs.H/ D lim!

n

Fredh.Hn/

and, without Kuiper’s theorem, prove in the same way that the set of connected com-ponents of the space of sections of this bundle is isomorphic to K.A/.X/.

There is an obvious ring homomorphism (since the Hilbert tensor product H ˝His isomorphic to H )

K ˝K �� K

If A and A0 are bundles of algebras on X modelled on K , we may use this homo-morphism to get a new algebra bundle on X , which we denote by A˝A0. From thecocycle point of view, we have a commutative diagram, where the top arrow is inducedby complex multiplication

S1 � S1

��

�� S1

��

U.H/ � U.H/ ��

��

U.H ˝H/

��

PU.H/ � PU.H/ �� PU.H ˝H/.It follows that W3.A˝A0/ D W3.A/C W3.A0/ in H 3.X IZ/ and one gets a “cup-product”

K.˛/.X/ �K.˛0/.X/ �� K.˛C˛0/.X/.

(well defined up to non canonical isomorphism: see 2.1). This is a particular case of a“graded cup-product” which will be introduced in the next section.

3 Graded twisted K -theory in the finite and infinite-dimensionalcases

We are going to change our point of view and now consider Z=2-graded finite-dimen-sional algebras which are central and simple (in the graded sense). We are only inter-

126 M. Karoubi

ested in the complex case. The real case is treated with great details in [19] and doesnot seem to generalize in the infinite-dimensional framework 11 .

In the complex case, there are just two “types” of such graded algebras (up to Moritaequivalence12) which are R D C and C�C D CŒx�=.x2�1/. For a typeR of algebra,the graded inner automorphisms of A D R y End.V0˚ V1/may be given by either anelement of degree 0 or an element of degree 1 in A�. This gives us an augmentation(whose kernel is denoted by Aut0.A//:

Aut.A/ �� Z=2.

Therefore, for bundles of Z=2-graded algebras modelled on A, we already get aninvariant in H 1.X IZ=2/, called the “orientation” of A and which may be representedby a real line bundle. A typical example is the (complexified) Clifford bundle C.V /,associated to a real vector bundle V of rank n. Its orientation invariant is the firstStiefel–Whitney class associated to V (cf. [19]). Note that the type R of C.V / is Cif n is even and CŒx�=.x2 � 1/ Š C �C if n is odd13.

For the second invariant, let us start with M2n.C/ as a basic graded algebra to fixthe ideas and let us put a C*-algebra metric onA. We have an exact sequence of groupsas in the ungraded case (where PU0.2n/ D PU.2n/ \ Aut0.A//

1 �� S1 �� U.n/ � U.n/ �� PU0.2n/ �� 1.

Therefore, a bundle with structural group PU0.2n/ also has a class in H 2.X IS1/ DH 3.X IZ/ which is easily seen to have order n as in the ungraded case. A similarargument holds if the graded algebra is Mn.C/ �Mn.C/. It follows that the “gradedBrauer group” GBr.X/ is

GBr.X/ D Z=2 �H 1.X IZ=2/ � Tors.H 3.X IZ//as already quoted in the introduction (see [19] for the explicit group law on GBr.X/). IfA is a bundle of Z=2-graded finite-dimensional algebras, we define the graded twistedK-theoryKA.X/ as the graded K-theory of the graded algebra �.X;A/ as recalled inSection 1. This definition only depends on the class of A in GBr.X/. We recover thedefinition in [19] by using again the Serre–Swan theorem as in 2.1. For instance, if weconsider the bundle of (complex) Clifford algebras C.V /, associated to a real vectorbundle V , the invariants we get are w1.V / and W3.V /, as quoted in the introduction.If these invariants are trivial, the bundle V is cspinorial of even rank and C.V /may be

11This is not quite true if we work in the context of “Real”K-theory in the sense of Atiyah [2]. We shall notconsider this generalization here, although it looks interesting in the light of “equivariant twistedK-theory”as we shall show in §6.

12This means that we are allowed to take the graded tensor product with End.V0˚V1/ with the obviousgrading.

13If V is oriented and even dimensional for instance, this does not imply that C.V / is a bundle of gradedalgebras of typeM2.ƒ/ for a certain bundle of ungraded algebrasƒ. However, for suitable vector bundlesV0 and V1, this is the case for the graded tensor product C.V / y End.V0 ˚ V1/.


identified with the bundle of endomorphisms End.SC ˚ S�/, where SC and S� arethe even and odd “spinors” associated to the Spinc-structure.

In order to define graded twisted K-theory in the infinite-dimensional case, wefollow the same pattern as in §2. For instance, let us take a graded bundle of algebras A

modelled on M2.K/: it has 2 invariants, one in H 1.X IZ=2/, the other in H 3.X IZ/(and not just in the torsion part of this group). We then define KA.X/ as the gradedK-theory of the graded algebra �.X;A/, according to §1. The same definition holdsfor bundles of graded algebras modelled on K �K DKŒx�=.x2 � 1/ DK y Kx.

If C 0;1 is the Clifford algebra C � C with its usual graded structure, the generalresults of §1 show that the group KA.X/ fits into an exact sequence:

K.A yC0;1/1 .X/ �� K

.A/1 .X/ �� KA.X/ �� K.A

yC0;1/.X/ �� K.A/.X/

where K.A/i .X/ denotes in general the Ki -group of the Banach algebra �.X;A/.Let us now assume that A is oriented modelled on M2.K/ (which means that the

structural group of A may be reduced to PU0.H ˚H/; see below or 3.2 in the finite-dimensional situation). We are going to show that A may be written as M2.A

0/, withthe obvious grading, A0 being an ungraded bundle of algebras modelled on K . For thispurpose, we write the commutative diagram

S1

��

�� S1

��

U.H/ ��

��

U.H/ � U.H/

��

PU.H/ �� PU0.H ˚H/where the first horizontal map is the identity and the others are induced by the diagonal.This shows that H 1.X IPU.H// � H 1.X IPU0.H ˚ H//, which is equivalent tosaying that A may be written as M2.A

0/ for a certain bundle of algebras A0.Therefore, A y C 0;1 is Morita equivalent to A0�A0 andKA.X/ is theK-theory of

the ring homomorphism (more precisely the functor defined by the associated extensionof the scalars, as we shall consider in other situations)

A0 �A0 �� M2.A0/

defined by .a; b/ 7! �a 00 b

�(no grading). We should also note that A0 is Morita equiv-

alent to A as an ungraded bundle of algebras. Since the usual K-theory (resp. gradedK-theory) is invariant under Morita equivalence (resp. graded Morita equivalence), theprevious considerations lead to the following theorem:

Theorem 3.1. Let A be an oriented bundle of graded algebras modelled on M2.K/.Then KA.X/ is isomorphic to K.A/.X/ via the identification above.

128 M. Karoubi

The same method may be applied when A is an oriented bundle of graded algebrasmodelled on K�K DKŒx�=.x2�1/. The graded oriented automorphisms of K�K

induced by PU.H ˚H/ are diagonal matrices of the type

�a 0

0 a

�:

This shows that A is isomorphic to A0 �A0 and A y C 0;1 is isomorphic to M2.A0/.

Therefore, KA.X/ is the Grothendieck group of the ring homomorphism A0 !M2.A

0/, defined by

a 7!�a 0

0 a

�:

Hence we have the following theorem, analogous to 3.5:

Theorem 3.2. Let A be an oriented bundle of graded algebras modelled on K �K .Then A is isomorphic to A0 �A0 and the group KA.X/ is isomorphic to K1.A0/ viathe identification above.

Remark 3.3. One may notice that if A is a bundle of oriented graded algebras modelledon M2.K/, the associated bundle with fiber M2.K

C/14 has a section " of degree 0and of square 1, which commutes (resp. anticommutes) with the elements of degree 0(resp. 1); it is simply defined by the matrix

�1 0

0 �1�:

Finally, we would like to give an equivalent definition of KA.X/ in terms of Fred-holm operators as in §2. This was done in [19] if the class of A belongs to the torsiongroup of H 3.X IZ/ and in [8] for the general case. We shall give another treatmenthere, using again the K-theory of graded Banach algebras.

Following the general notations of §2, we have an exact sequence of bundles ofgraded Banach algebras

0 �� A �� B �� B=A �� 0.

Since the graded K-groups of B are trivial, we see as in 2.6 that GrK.�.X;A// isisomorphic to GrK1.�.X;B=A//.

In order to shorten the notations, we denote by B the graded Banach algebra�.X;B/, by ƒ the graded Banach algebra �.X;B=A/ and by f the surjective mapB ! ƒ. The following lemma15 may be proved in the same way as in [29], p. 78:

14KC is the ring K which a unit added15It might be helpful for a better understanding to notice that the category of finitely generated free

End.H/-modules is equivalent to the category of Hilbert spaces Hn for n 2 N. This “local” situation istwisted by the group PU.H/.


Lemma 3.4. Any element of GrK1.ƒ/ may be written as the homotopy class of apair .E; "/ where E is a free graded B-module and " is a grading of degree 1 of theassociated ƒ-module (see 1.4 for the definition of a grading).

By the well-known dictionary between modules and bundle theory, we may view " asa grading of a suitable bundle of free End.H/=K-modules. By spectral theory, we mayalso assume that " is self-adjoint. Finally, following [29], we define a quasi-grading16

of E as a family of Fredholm endomorphisms D such that

1. D� D D,

2. D is of degree 1.

The following theorem is the analogue in the graded case of Theorem 2.8 (cf [29],p. 78/79).

Theorem 3.5. The (graded) twisted K-group KA.X/ is the Grothendieck group as-sociated to the semi-group of homotopy classes of pairs .E;D/ where E is a freeZ=2-graded B-module andD is a family of Fredholm endomorphisms of E which areself-adjoint and of degree 1.

Remark 3.6. Let us assume that A is oriented modelled on M2.K/. The descriptionabove gives a Fredholm description of GrK1.A/ D K1.A/: we just take the homo-topy classes of sections of the associated bundle of self-adjoint Fredholm operatorsFredh�.B/whose essential spectrum is divided into two non empty parts17 in RC� andR��.

This Fredholm description of KA.X/ enables us to define a cup-product

KA.X/ �KA0

.X/ �� KA yA0

.X/

where A y A0 denotes the graded tensor product of A and A0. This cup-product isgiven by the same formula as in [19], p. 19, and generalizes it:

.E;D/Y .E 0;D0/ D .E y E 0;D y 1C 1 y D0/(see the appendix about the origin of this formula in usual topological K-theory).

To conclude this section, let us consider a locally compact spaceX and a bundle ofgraded algebras A on X . For technical reasons, we assume the existence of a compactspace Z containing X as an open subset, such that A extends to a bundle (also calledA) on Z. There is an obvious definition of KA.X/ as a relative term in the followingexact sequence (where T D Z �X and A0 D A y C 0;1):

KA0

.Z/ �� KA0

.T / �� KA.X/ �� KA.Z/ �� KA.T /.

16“quasi-graduation” in French.17See the appendix about the role of self-adjoint Fredholm operators inK-theory.

130 M. Karoubi

By the usual excision theorem in topologicalK-theory, one may prove that this defini-tion of KA.X/ is independent from the choice of Z.

The method described before and also in [29], §3, shows how to generalize thedefinition ofKA.X/ in this case: one takes homotopy classes of pairs .E;D/ as in 3.12,with the added assumption that the familyD is acyclic at infinity. In other words, thereis a compact setS � X , such thatDx is an isomorphism whenx … S (see [29], p. 89–97for the technical details of this approach). This Fredholm description of KA.X/ willbe important in the next section for the description of the Thom isomorphism.

4 The Thom isomorphism in twisted K -theory18

Let V be a finite-dimensional real vector bundle on a locally compact space X whichextends over a compactification of X as was assumed in 3.15. Then the complexifiedClifford bundle C.V / has a well-defined class in the graded Brauer group of X . If A

is another twist on X , we can consider the graded tensor product A y C.V / and theassociated group KA yC.V /.X/. As was described in 3.15, it is the group19 associatedto pairs .E;D/ where E is a graded bundle of free B-modules and D is a family ofFredholm endomorphisms which are of degree 1, self-adjoint and acyclic at 1. Letus now consider the projection � W V ! X . For simplicity’s sake, we shall often callE;A;B; : : : the respective pull-backs of E;A;B; : : : via this projection. Since B

and C.V / are subbundles of B y C.V /, E may be provided with the induced B andC.V /-modules structures.

Theorem 4.1. Let d.E;D/be an element of KA yC.V /.X/ with the notations above.We define an element t .d.E;D// in the groupKA.V / as d.��.E/;D0/whereD0 is thefamily of Fredholm operators on ��.E/, defined over the point v in V (with projectionx on X ) as

D0.x;v/ D Dx C �.v/where �.v/ denotes the action of the element v of the vector bundle V considered as asubbundle of C.V /. The homomorphism

t W KA yC.V /.X/! KA.V /

(t for “Thom”) defined above is then an isomorphism.

Proof. 20 We should first notice thatV may be identified with the open unit ball bundle ofthe vector bundle V and is therefore an open subset of the closed unit ball bundle of V .Moreover, since V and A extends to a compactification of X , the required conditionsin 3.15 for the definition of the twisted K-theory of X and V are fulfilled.

18See also [16].19We should note thatE is a B-module, not an A-module. Nevertheless, we shall keep the notationKA.20According to a suggestion of J. Rosenberg, it should be possible to give a proof with the KK-theory

of Kasparov by describing an explicit inverse to the homomorphism t . However, KK-theory is out of thescope of this paper.


We shall now provide two different proofs of the theorem.The first one, more elementary in spirit, consists in using a Mayer–Vietoris argument

which we can apply here since the two sides of the formula above behave as cohomologytheories21 with respect to the base X . Therefore, we may assume that A and V aretrivial: this is a special case of the theorem stated in [30], pp. 211/212.

The second one is more subtle and may be generalized to the equivariant case. Letus first describe the Thom isomorphism for complex V . This is a slight modificationof Atiyah’s argument using the elliptic Dolbeault complex [3]. More precisely, weconsider the composite map

� W KA.X/! KA yC.V /.X/ t! KA.V /:

The first map � is the cup-product with the algebraic “Thom class” which is ƒV pro-vided with the classical Clifford graded module structure. This map is an isomorphismfrom well-known algebraic considerations (Morita equivalence). Therefore t is anisomorphism if and only if � is an isomorphism. On the other hand, � is just thecup-product with the topological Thom class TV which belongs to the usual topolog-ical K-theory K.V / of Atiyah and Hirzebruch [5], [6]. In order to prove that � is anisomorphism, we may now use the exact sequence in 3.3 to reduce ourselves to theungraded twisted case. In other words, it is enough to show that the cup-product withTV induces an isomorphism

‰ W K.A/.X/! K.A/.V /:

In order to prove this last point, Atiyah defines a reverse map22

‰0 W K.A/.V /! K.A/.X/:

He shows that ‰0‰ D Id and, by an ingenious argument, deduces that ‰‰0 D Id aswell.

Now, as soon as Theorem 4.2 is proved for complex V , the general case followsfrom a trick already used in [30], p. 241: we consider the following three Thom homo-morphisms which behave “transitively”:

KA yC.V / yC.V / yC.V /.X/ �� KA yC.V / yC.V /.V /

�� KA yC.V /.V ˚ V / �� KA.V ˚ V ˚ V /.We know that the composites of two consecutive arrows are isomorphisms since V ˚Vcarries a complex structure. It follows that the first arrow is an isomorphism, which isessentially the theorem stated (using Morita equivalence again).

21Strictly speaking, one has to “derive” the two members of the formula, which can be done easily sincethey are Grothendieck groups of graded Banach categories.

22More precisely, one has to define an index map parametrized by a Banach bundle, which is also classical[20].

132 M. Karoubi

Let A be a graded twist. As we have seen before, it has two invariants inH 1.X IZ=2/and inH 3.X IZ/. The first one provides a line bundle L in such a way that the gradedtensor product A1 D A y C.L/ is oriented. From Thom isomorphism and the consid-erations in 3.5/7, we deduce the following theorem which gives the relation betweenthe ungraded and graded twisted K-groups.

Theorem 4.2. Let A be a graded twist of type M2.K/ and A1 D A y C.L/, whereL is the orientation bundle of A. Then A1 may be written as A2 � A2 where A2 isungraded of type K . Therefore, we have the following isomorphisms

KA.X/ Š KA yC.L/ yC.L/.X/ Š KA1.L/ Š K.A2/1 .L/:

Let A be a graded twist of type K�K . With the same notations, we have the followingisomorphisms

KA.X/ Š KA yC.L/ yC.L/.X/ Š KA1.L/ Š K.A1/.L/:

5 General equivariant K -theory

Note: this section is inserted here for the convenience of the reader as an introductionto §6. It is mainly a summary of results found in [39], [12], [35] and [8].

LetA be a Banach algebra andG a compact Lie group acting onA via a continuousgroup homomorphism

G �� Aut.A/

where Aut.A/ is provided with the norm topology. We are interested in the categorywhose objects are finitely generated projectiveA-modulesE together with a continuousleft action of G on E such that we have the following identity holds (g 2 G, a 2 A,e 2 E)

g:.a:e/ D .g:a/:.g:e/with an obvious definition of the dots. We define KG.A/ as the Grothendieck groupof this category C D PG.A/. By the well-known dictionary between bundles andmodules, we recover the usual equivariant K-theory defined by Atiyah and Segal [39]ifA is the ring of continuous maps on a compact spaceX (thanks to the lemma below).It may also be defined as a suitable semi-direct product of G by A (cf. [12]).

More generally, if A is a Z=2-graded algebra (where G acts by degree 0 automor-phisms), we define the graded equivariantK-theory ofA, denoted by GrKG.A/, as theGrothendieck group of the forgetful functor

C0;1 �� C

where C0;1 is the category of “graded objects” in PG.A/ which is defined asPG.A y C 0;1/.

In the same spirit as in §1, we define “derived” groupsKp;qG .A/ as GrKG.A yCp;q).They satisfy the same formal properties as the usual groups Kn.A/ (also denoted by


K�n.A/ with n D q � p), for instance Bott periodicity. The following key lemmaenables us to translate many general theorems of K-theory into the equivariant frame-work:

Lemma 5.1. Let E be an object of the category PG.A/. Then E is a direct summandof an object of the type A˝C M where M is a finite-dimensional G-module.

Proof. ([39], p. 134). Let us consider the union � of all finite-dimensional invariantsubspaces of theG-Banach spaceE. According to a version of the Peter-Weyl theoremquoted in [39], this union is dense inA. We now consider a set e1; : : : ; en of generatorsof E as an A-module. Since � is dense in A and E is projective, one may choosethese generators in the subspace � . Let M1; : : : ;Mn be finite-dimensional invariantsubspaces of E containing e1; : : : ; en respectively and let M be the following directsum:

M DM1 ˚ � � � ˚Mn

We define an equivariant surjection

� W A˝C M Š .A˝C M1/˚ � � � ˚ .A˝C Mn/! E:

between projective left A-modules by the formula

�.1 ˝m1; : : : ; n ˝mn/ D 1m1 C � � � C nmn:This surjection admits a section, which we can average out thanks to a Haar measurein order to make it equivariant. Therefore, E is a direct summand in A˝C M as statedin the lemma.

As for usual K-theory, one may also define equivariant K-theory for non unitalrings and, using Lemma 5.2 above, prove that any equivariant exact sequence of ringson which G acts

0 �� A0 �� A �� A00 �� 0

induces an exact sequence of equivariant K-groups

Kn�1G .A/ �� Kn�1G .A00/ �� KnG.A0/ �� KnG.A/

�� KnG.A00/

and an analogous exact sequence in the graded equivariant framework.After these generalities, let us assume that G acts on a compact space X and let A

be a bundle of algebras modelled on K . We define the (ungraded) equivariant twistedK-groupK.A/G .X/ asKG.A/, where A is the Banach algebra of sections of the bundleA. Similarly, if A is a bundle of graded algebras modelled on K �K or M2.K/, wedefine the (graded) equivariant twisted K-group KA

G .X/ as GrKG.A/, where A is thegraded Banach algebra of sections of A.

As seen in §3, it is also natural to consider free graded B-modules together with acontinuous action of G (compatible with the action on X ) and a family of FredholmoperatorsD which are self-adjoint of degree 1, commuting with the action of G. With

134 M. Karoubi

the same ideas as in §3, we can show that the Grothendieck group of this category isisomorphic to KA

G .X/. This is essentially the definition proposed in [8].An example was in fact given at the beginning of the history of twisted K-theory.

In [19], §8, we defined a “power operation”

P W KA.X/! KAyn

Sn.X/

where Sn denotes the symmetric group on n letters acting on Ayn (in the spirit of

Atiyah’s paper on power operations [2]). According to the general philosophy ofAdamsand Atiyah, one can deduce from this nth power operation new “Adams operations”(n being odd and the group law in GBr.X/ being written multiplicatively):

‰n W K˛.X/! K˛n

.X/˝Z n

where ˛ belongs to Z=2�H 1.X IZ=2/�H 3.X IZ/ andn denotes the free Z-modulegenerated by the nth roots of unity in C (the ring of cyclotomic integers23). One canprove, following [19], that these additive maps ‰n satisfy all the required propertiesproved by Adams. We shall come back to them in §7, making the link with operationsrecently defined by Atiyah and Segal [9].

In order to fix ideas, let us consider the graded version of equivariant twisted K-theory KA

G .X/, where A is modelled on C � C with the obvious grading. We haveagain a “Thom isomorphism”:

t W KA yC.V /G .X/! KA

G .V /

whose proof is the same as in the non equivariant case (using elliptic operators). Onthe other hand, if L is the orientation line bundle of A, the group G acts on L in a waycompatible with the action onX . Therefore, A1 D A y C.L/ is an algebra bundle withtrivial orientation. and A1 yC 0;1 is naturally isomorphic to A1�A1 D A1Œx�=.x

2�1/.As in 4.4, this shows that the graded twisted group KA

G .X/ is naturally isomorphic to

the ungraded twisted K-group KA1G .L/. The same method shows that we can reduce

graded twistedK-groups to ungraded ones if A is a bundle of graded algebras modelledon M2.K/.

Let us mention finally one of the main contributions of Atiyah and Segal to thesubject ([8], §6, see also [17]), which we interpret in our language. One is interested inalgebra bundles A on X modelled on K , provided with a left G-action. The isomor-phism classes of such bundles are in bijective correspondence with principal bundlesP over PU.H/ (acting on the right) together with a left action of G.

Such an algebra bundle T is called “trivial” if T may be written as the bundle ofalgebras of compact operators in End.V /, where V is aG-Hilbert bundle. Equivalently,this means that the structure group of T can be lifted equivariantly to U.H/ (in a waycompatible with the G-action).

23The introduction of this ring is absolutely necessary in the graded case.


We now say that two such algebra bundles A and A0 are equivalent if there existtwo trivial algebra bundles T and T 00 such that A ˝ T and A0 ˝ T 00 are isomorphicas G-bundles of algebras. The quotient is a group since the dual of A is its inverse viathe tensor product of principal PU.H/-bundles. This is the “equivariant Brauer group”BrG.X/.

A closely related definition (in the framework of C*-algebras and for a locallycompact group G) is given in [17]. It is very likely that it coincides with this onefor compact Lie groups, in the light of an interesting filtration described in this paper,probably associated to a spectral sequence.

Theorem 5.2 (cf. [8], Proposition 6.3). Let XG D EG �G X be the Borel spaceassociated to X . Then the natural map

BrG.X/! Br.XG/

is an isomorphism.

The interest of this theorem lies in the fact that the equivariant K-theoriesKG.�.X;A// and KG.�.X;A0// are isomorphic if A and A0 are equivalent. Thisfollows from the well-known Morita invariance in operator K-theory. We shall studyconcrete applications of this principle in the next section.

6 Some computations of twisted equivariant K -groups

Let us look at the particular case of the ungraded twistedK-groupsK.A/G .X/ where Gis a finite group acting on the trivial bundle of algebras A D X �Mn.C/ via a grouphomomorphism G ! PU.n/.24 We define zG as the pull-back diagram

zG ��

��

SU.n/

��

G �� PU.n/

Therefore, zG is a central covering of G with fiber �n (whose elements are denoted byGreek letters such as ). The following definition is already present in [31], §2.5 (forn D 2):

Definition 6.1. A finite-dimensional representation� of zG is of “linear type” if�.u/ D�.u/ for any 2 �n. We now consider the category EA

QG .X/l whose objects arezG-A-modules as before, except that we request that the zG-action be of linear type andcommutes with the action of A. By Morita invariance, EA

QG .X/l is equivalent to the

category E QG.X/l of finite-dimensional zG-bundles on X , the action of zG on the fibersbeing of linear type.

24However, we don’t assume thatG acts trivially onX in general.

136 M. Karoubi

Theorem 6.2.25 The (ungraded) twistedK-theoryK.A/G .X/ is canonically isomorphicto the Grothendieck group of the category E zG.X/l .

Proof. One just repeats the argument in the proof of Theorem 2.6 in [31], where A is aClifford algebra C.V / and Z=2 plays the role of �n. We simply “untwist” the actionof zG thanks to the formula (F) written explicitly in the proof of 2.6 (loc. cit.).

For A D X � A with A D Mn.C/, the previous argument shows that K.A/G .X/ isa subgroup of the usual equivariant K-theory K zG.X/. From now on, we shall write

KAG.X/ instead of K.A/G .X/. Similarly, in the graded case (A D Mn.C/ �Mn.C/ orM2n.C/), we shall write KAG.X/ instead of KA

G .X/. If X is a point and G is finite,

K.A/G .X/ is just the K-theory of the semi-direct product G Ë A.

Theorem 6.3. Let G be a finite group acting on the algebra of matrices A D Mn.C/and let zG be the central extension G by �n described in 6.1. Then, for X reduced to apoint, the group K.A/G .X/ D K.G Ë A/ is a free abelian group of rank the number ofconjugacy classes in G which split into n conjugacy classes in zG.

Proof. We can apply the same techniques as the ones detailed in [31], §2.6/12 (forn D 2). By the theory of characters on zG, one is looking for functions f on zG (whichwe call of “linear type”) such that

1. f .hgh�1/ D f .g/,2. f .x/ D f .x/ if is an nth root of the unity.

The C-vector space of such functions is in bijective correspondence with the space offunctions on the set of conjugacy classes of G which split into n conjugacy classesof zG.

Like the Brauer group of a space X , one may define in a similar way the Brauergroup Br.G/ of a finite group G by considering algebras A DMn.C/ as above with aG-action (see [24] for a broader perspective ; this is also a special case of the generaltheory of Atiyah and Segal mentioned at the end of §5). From the diagram written in6.1, one deduces a cohomology invariant

w2.A/ 2 H 2.GI�n/and (via the Bockstein homomorphism) a second invariant W3.A/ 2 H 2.GIS1/ DH 2.GIQ=Z/ D H 3.GIZ/. It is easy to show that this correspondence induces awell-defined map

W3 W Br.G/! H 3.GIZ/:The following theorem is a special case of 5.9 in a more algebraic situation.

25There is an obvious generalization when A is infinite-dimensional. However, for our computations, werestrict ourselves to the finite-dimensional case.


Theorem 6.4. Let G be a finite group. Then the previous homomorphism

W3 W Br.G/! H 3.GIZ/

is bijective.

Proof. First of all, we remark that H 3.GIZ/ Š H 2.GIQ=Z/ is the direct limit ofthe groups H 2.GI�m/ through the maps H 2.GI�m/! H 2.GI�p/ when m dividesp. This stabilization process corresponds on the level of algebras to the tensor productA 7! A ˝ End.V /, where V is a G-vector space of dimension p=m. Therefore, themap W3 is injective.

The proof of the surjectivity is a little bit more delicate (see also 5.9). We can sayfirst that it is a particular case of a much more general result proved by A. Fröhlich andC. T. C. Wall [24] about the equivariant Brauer group of an arbitrary field k: there is asplit exact sequence (with their notations)

0 �� Br.k/ �� BM.k;G/ �� H 2.GIU.k// �� 0

where Br.k/ is the usual Brauer group of k, U.k/ is the group of invertible elements ink and BM.k;G/ is a group built out of central simple algebras over k with aG-action.Since Br.C/ D 0 and H 2.GIU.k// D H 2.GIQ=Z/, the theorem is an immediateconsequence.

Here is an elementary proof suggested by the referee (for k D C). If we start witha central extension

1 �� n �� zG �� G �� 1

we consider the finite dimensional vector space

H D ff 2 L2. zG/ such that f .�/ D �1f .�/ with .; �/ 2 �n � zGg:

Then zG acts by left translation on H in such a way that this action is of linear typeas described in 6.2. Therefore, we have a projective representation of G on H andEnd.H/ is a matrix algebra where G acts.

Remark 6.5. Let us consider an arbitrary central extension zG of G by �n Š Z=nassociated to a cohomology class c 2 H 2.GIZ=n/. We are interested in the set ofelements g ofG such that the conjugacy class hgi splits into n conjugacy classes in zG1.This set only depends on the image of c inH 3.GIZ/ via the Bockstein homomorphismH 2.GIZ=n/ ! H 3.GIZ/ (cf. 6.6). In other words, two central extensions of G byZ=n with the same associated image by the Bockstein homomorphism have the sameset of n-split conjugacy classes.

138 M. Karoubi

In order to show this fact, let us consider the following diagram:

�n ��

��

�nm

��

zG1 ��

�

��

zG

��

G G,

and an element g1 of zG1. The conjugacy class of g D �.g1/ splits into n conjugacyclasses in zG1 if and only if there is a trace function f on zG1 with values in C such thatf .g1c/ D f .g1/c when c 2 �n. Such a trace function extends obviously to zG, whichyields to the result, since the direct limit of the groups H 2.GIZ=nm/ is preciselyH 2.GIQ=Z/ Š H 3.GIZ/.

This remark may be generalized as follows according to a suggestion of J.-P. Serre:let zG1 and zG be two group extensions (not necessary central) of G by abelian groupsC1 and C of ordersm1 andm respectively, such that the following diagram commutes(with ˛ injective):

C1˛ ��

��

C

��

zG1 ��

�

��

zG

��

G G.

The previous argument shows that if an element g ofG splits intom conjugacy classesin zG, it splits intom1 conjugacy classes in zG1: take trace functions f on zG with valuesin C such that f .gc/ D f .g/c (we write multiplicatively the abelian group C ). Theconverse is true if the extension of G by C is central.

Let us now assume thatX is not reduced to a point. We can use the Baum–Connes–Kuhn–Słominska Chern character [11], [33], [41] which is defined onK�.X/ (for anyfinite group �), with values in the direct sum

Lh�iH even.Xg/C.�/. In this formula,

h�i runs through all the conjugacy classes of G, C.�/ being the centralizer of � (thecohomology is taken with complex coefficients). One of the main features of this“Chern character”

K�.X/ ��Lh�iH even.X� /C.�/

is the isomorphism it induces betweenK�.X/˝Z C and the cohomology with complexcoefficients on the right-hand side. If E is a �-vector bundle, the map is definedexplicitly by Formula 1.13, p. 170 in [11].

Let us now take for � the group zG previously considered and let us analyze theformula in this case. We shall view the right-hand side not just as a function on the


set of conjugacy classes h�i, but as a function f on the full group � with certain extraproperties which we shall explain now.

If we replace � by � 0 such that � 0 D ��1, we have a canonical isomorphism�� W H�.X� 0

/C.�0/ ! H�.X� /C.�/ induced by x 7! �x. This map exchanges f .�/

and f .� 0/ and we have the relation f .�/ D ��.f .� 0//, which says that f is essentiallya function on the set of conjugacy classes.

On the other hand, if the action of zG is of linear type, we have an extra relation, aneasy consequence of the formula in [11], which is f .��/ D �f .�/ when � is an nth

root of unity. To summarize, we get the following theorem.

Theorem 6.6. Let G be a finite group and A D Mn.C/ with a G-action. Thenthe ungraded twisted equivariant K-theory K.A/G .X/ is a subgroup of the equivariantK-theory K zG.X/, where zG is the pull-back diagram

zG ��

�

��

SU.n/

��

G �� PU.n/.

More precisely,K.A/G .X/˝Z C may be identified with the C-vector space of functionsf on � D zG with f .�/ in Heven.Xg/C.�/, �.�/ D g, such that the following twoidentities hold:

1. If � 0 D ��1, one has f .�/ D ��.f .� 0//, according to the formula above.

2. f .��/ D �f .�/ if � is an nth root of unity.

In particular, if X is reduced to a point, we have �� D Id and K.A/G .X/ is free withrank the number of conjugacy classes of G which split into n conjugacy classes in zG(as seen in 6.5.)

Remark 6.7. This theorem is not really new. In a closely related context, one findssimilar results in [1] and [42]. We should also notice that the same ideas have beenused in [31] for representations of “linear type”. Finally, the theorem easily extendsto locally compact spaces if we consider cohomology with compact supports on theright-hand side.

Theorem 6.8. Let A be any finite-dimensional graded semi-simple complex algebrawith a graded action of a finite group G. Then the graded K-theory GrK0.A0/ ˚GrK1.A0/ of the semi-direct product A0 D G Ë A is a non trivial free Z-module. Inparticular, if V is a real finite-dimensional vector space with a G action, the groupKAG.V /˚KAG.V ˚ 1/ is free non trivial thanks to the Thom isomorphism.

Proof. The algebraGËA is graded semi-simple over the complex numbers. Therefore,it is a direct sum of graded algebras Morita equivalent toMn.C/�Mn.C/ orM2n.C/.In both cases, the graded K-theory is non trivial. The last part of the theorem followsfrom 4.2.

140 M. Karoubi

The following theorem is a direct consequence of the previous considerations:

Theorem 6.9. Let us now assume thatA DM2n.C/ isG-oriented as a graded algebra:in other words, there is an involutive element " of A of degree 0 which commutes withthe action of G and commutes (resp. anticommutes) with the elements of A of degree 0(resp. 1). Then, the gradedK-theory GrK�.A0/, withA0 D GËA, is a finitely generatedfree module concentrated in degree 0. More precisely, one has GrK0.A0/ D K.A0/and GrK1.A0/ D 0. In particular, if V is an even-dimensional real vector space and ifA y C.V / is G-oriented, we have (via the Thom isomorphism)

KAG.V / D KA yC.V /G .P / D K.G Ë A y C.V // and KAG.V ˚ 1/ D 0where P is a point. If we write A y C.V / as an algebra of matrices Mr.C/ with arepresentation � of G and call zG the associated central extension by �r , the rank ofKAG.V / is the number of conjugacy classes of G which split into r conjugacy classesin zG.

In the abelian case, the following two theorems are related to results obtained byP. Hu and I. Kriz [26], using different methods.

Theorem 6.10. Let us consider the algebra A D Mn.C/ provided with an action ofan abelian group G and P a point. Then the ungraded twistedK-theoryK.A/G .P /� DK�.A0/, with A0 D G Ë A, is concentrated in degree 0 and is a free Z-module. If wetensor this group with the rationals and if we look at it as an R.G/ ˝ Q D QŒG�-module, it may be identified with R.G0/ ˝ Q for a suitable subgroup G0 of G. Inparticular, the rank of K0.A0/ divides the order of G.

Proof. The first part of the theorem is a consequence of the previous more generalconsiderations. As we have shown before, the algebra A0 gives rise to the followingcommutative diagram:

zGn�

��

�� SU.n/

��

G �� PU.n/,

the fibers of the vertical maps being�n. The subset of elements Qg in zGn such that �. Qg/splits into n conjugacy classes is just the center Z. zGn/ of zGn (since G is abelian).Let us put �n D �.Z. zGn//. Then K.A0/ may be written as KAG.P / where P is apoint. According to Theorem 6.10, this is the subgroup of the representation ring ofzGn generated by representations of linear type. At this stage, it is convenient to maken D1 by extension of the roots of unity, so that we have an extension of G by Q=Z

Q=Z �� zG �� G

(the “linear type” finite-dimensional representations of zG are the same as the originallinear type finite-dimensional representations of zGn). We call R. zG/l/ the associated


Grothendieck group. By the theory of characters, we see thatR. zG/l˝Q is isomorphicto R.Z. zG//l ˝Q, since the characters of such linear type representations of zG vanishoutside Z. zG/. On the other hand, if we denote by G0 the image of Z. zG/ in G, theextension of abelian groups

Q=Z �� Z. zG/ �� G0

splits (non canonically). This means that we can identify R. zG/l ˝Q with the repre-sentation ring R.G0/ ˝ Q as an R.G/ ˝ Q-module. This proves the last part of thetheorem.

Theorem 6.11. Let us consider a graded algebra A D M2n.C/ provided with a nonoriented action of an abelian group G (with respect to the grading). Then the gradedtwistedK-theoryKAG.P /� D GrK�.A0/, with A0 D G ËA, is concentrated in a singledegree (0 or 1) and is a free Z-module. If we tensor this group with the complexnumbers and if we look at it as an R.G/ ˝ C D CŒG�-module, it may be identifiedwith R.G0/˝Q for a suitable subgroup G0 of G. In particular, the rank of KAG.P /�divides the order of G.

Proof. Let L be the orientation bundle of A (with respect to the action of G). If wechange A into A y C.L/ and if we apply the Thom isomorphism theorem, we have tocompute KAG.L/� where G acts on A (resp. L) in an oriented way (resp. non orientedway).

Let us apply Theorem 6.10 in this situation: since G is abelian, the function f ofthe theorem must be equal to 0 on the elements of zG which are not inZ. zG/. Therefore,the relevant group KAG.L/� is reduced to KG0.L/ (after tensoring with C and whereG0 is the image of Z.G/ in G). We now consider two cases:

1. the action � of G0 on L is oriented, in which case we only find R.G0/ (with ashift of dimension). Therefore, the dimension of KAG.P /˝C is the order of G0which divides the order of G.

2. the action � ofG0 on L is not oriented. We then find a direct sum of copies of C,each one corresponding to an element ofG0 such that �.g0/ D �1. The dimensionof KAG.L/˝C is therefore half the order of G0, hence divides jGj=2.

Remark 6.12. For an oriented action of G on M2n.C/, Theorem 3.5 enables us tosolve the analogous problem in ungraded twisted K-theory, which is done in 6.14.

On the other hand, as we already mentioned, the two last theorems are related toresults of P. Hu and I. Kriz [26].

In [31] we also perform analogous types of computations when A is a Cliffordalgebra and G is any finite group, again using the Thom isomorphism. For instance,if G is the symmetric group Sn acting on the Clifford algebra of Rn via the canonicalrepresentation ofSn on Rn, there is a nice relation betweenK-theory and the pentagonalidentity of Euler (cf. [31], p. 532).

142 M. Karoubi

We would like to point out also that the theoryK˙.X/ introduced recently byAtiyahand Hopkins [7] is a particular case of twisted equivariant K-theory. As a matter offact, it was explicitly present in [28], §3, 40 years ago, before the formal introductionof twisted K-theory. Here is a detailed explanation of this identification.

According to [7], the definition of K˙.X/ (in the complex or real case) is thegroup KZ=2.X � R8/, where Z=2 acts on X and also on R8 D R � R7 by .; �/ 7!.�;�/. According to the Thom isomorphism in equivariant K-theory (proved inthe non spinorial case in [30]), it coincides with an explicit graded twisted K-groupKAZ=2.X/, as defined in [28]. Here A is the Clifford algebra C.R2/ D C 1;1 of R2

provided with the quadratic form x2 � y2 and where Z=2 acts via the involution.; �/ ! .�;�/ on R � R (this is also mentioned briefly in [7], p. 2, footnote 1).This identification is valid as well in the real framework, where we have 8-periodicity.

These groupsKAZ=2.X/ were considered in [28], §3.3, in a broader context: Amaybe any Clifford algebra bundle C.V / (where V is a real vector bundle provided witha non degenerate quadratic form) and Z=2 may be replaced by any compact Lie groupacting in a coherent way on X and V . The paper [31] gives a method to computethese equivariant twisted K-groups. As quoted in the appendix, the real and complexself-adjoint Fredholm descriptions (for the non twisted case) which play an importantrole in [7] were considered independently in [10] and [32].

7 Operations on twisted K -groups

This section is a partial synthesis of [19] and [9].Let us start with the simple case of bundles of (ungraded) infinite C*-algebras

modelled on K , like in [9]. As it was shown in [2] and [19], we have a nth power map

P W KA.X/! K.A˝n/Sn

.X/

where the symmetric group Sn acts on A˝n by permutation of the factors.

Lemma 7.1.26 The group K.A˝n/

Sn.X/ is isomorphic to the group K.A

˝n/0Sn

.X/ wherethe symbol 0 means that Sn is acting trivially on A˝n

Proof. As we have shown many times in §6, this “untwisting” of the action of thesymmetric group on A˝n is due to the following fact: the standard representation

Sn ! PU.H˝n/

can be lifted into a representation � W Sn ! U.H˝n/ in a way compatible with thediagonal action of elements of PU.H/, a fact which is obvious to check.

26We should note that this lemma is not true for KSn.A˝n/ for a general noncommutative ring A.

Therefore, it is not possible to define �-operations in this case. TwistedK-theory is somehow intermediarybetween the commutative case and the noncommutative one.


Remark 7.2. If we take a bundle of finite dimensional algebras modelled on A DEnd.E/ where E D Cr , there is another way to check (functorially) this untwisting ofthe action ofSn onA˝n: we identifyA˝n with End.E˝n/ and .A˝n/� with Aut.E˝n/.We have the following commutative diagram:

End.E˝n/�

�

��

D Aut.E˝n/

Sn ��

�

��End.E˝n/ D End.E/˝n.

The vertical map sends the invertible element ˛ to the automorphism .u 7! ˛u˛�1/.If � is a permutation, the horizontal map sends � to the automorphism

u1 ˝ � � � ˝ un 7! u�.1/ ˝ � � � ˝ u�.n/while the map � sends � to the automorphism of E˝n defined by

x1 ˝ � � � ˝ xn 7! x�.1/ ˝ � � � ˝ x�.n/:Finally, the composition �� , computed on a decomposable tensor of E˝n, gives therequired result:

x1 ˝ � � � ˝ xn � �� x�.1/ ˝ � � � ˝ x�.n/ u �� u1.x�.1//˝ � � � ˝ un.x�.n//

��1�� u�.1/.x1/˝ � � � ˝ u�.n/.xn/.

As was shown in [2], a Z-module map

R.Sn/ �� Z

defines an operation in twistedK-theory by taking the composite of the following maps:

K.A/.X/ �� K.A˝n/Sn

.X/ Š K.A˝n/0Sn

.X/ Š K.A/˝n.X/˝R.Sn/ �� K.A/˝n.X/.

This is essentially27 what was done in [9], §10, in order to define the n operation ofGrothendieck in this context for instance.

Let us call .F;r/ or even F (for short) a representative of the image of .E;D/ bythe composite of the maps (we now use the Fredholm description of twistedK-theory)

K.A/.X/ �� K.A˝n/Sn

.X/ �� K.A˝n/0An

.X/.

We can define the Adams operations ‰n with the method described in [19] (which weintend to generalize later on). For this, we restrict the action of Sn to the cyclic group

27The second homomorphism was not explicitly given however.

144 M. Karoubi

Z=n identified with the group of nth roots of the unity. Let us call Fr the subbundle ofF where the action of Z=n is given by !r , ! being a fixed primitive root of the unity.Then ‰n.E;D/ is defined by the following sum:

‰n.E;D/ Dn�1X0

Fr!r :

It belongs formally toK.A/˝n.X/˝Zn wheren is the ring ofn-cyclotomic integers.

However, if n is prime, using the action of the symmetric group Sn, it is easy to checkthat Fr is isomorphic to F1 if r ¤ 1. Therefore we end up in K.A/

˝n.X/, considered

as a subgroup of K.A/˝n.X/

Zn, as it was expected. It is essentially proved in [2]

that this definition of ‰n agrees with the classical one.There is another operation in twisted K-theory which is “complex conjugation”,

classically denoted by ‰�1, which maps K.A/.X/ to K.A/�1.X/ (if we write mul-

tiplicatively the group law in Br.X/). It is shown in [9], §10, how we can combinethis operation with the previous ones in order to get “internal” operations, i.e. mappingK.A/.X/ to itself.

It is more tricky to define operations in graded twisted K-theory. If ƒ is a Z=2-graded algebra, it is no longer true in general that a graded involution ofƒ is induced byan inner automorphism with an element of degree 0 and of order 2. A typical exampleis the Clifford algebra

.C ˚C/y2 D .C 0;1/ y2

which may be identified with the graded algebra M2.C/. If we put

e1 D�0 1

1 0

�and e2 D

�0 �ii 0

�

we see that there is no inner automorphism by an element of order 2 and degree 0permuting e1 and e2.

In order to define such operations in the graded case, we may proceed in at least twoways.We first remark that if A0 is an oriented bundle modelled on M2.K/, the groupsK.A

0/.X/ andKA0

.X/ are isomorphic (see 3.5). Moreover, the ungraded tensor productA0˝n is isomorphic to the graded one A0 yn (since M2.C/ y M2.C/ is isomorphic toM4.C/ D M2.C/ ˝M2.C//. Let us now assume that the bundle of algebras A ismodelled on K �K but not necessarily oriented and let L be the orientation real linebundle of A. Then A0 D A y C.L/ is oriented modelled on M2.K/ (C.V / denotesin general the Clifford bundle associated to V ). We may apply the previous method

to define operations from K.A0/.Y / D KA0

.Y / to KA0 yn.Y /. If we apply the Thom

isomorphism to the vector bundle Y D L with basis X (see §4), we deduce (for n

odd) operations fromKA.X/ toKAyn.X/. However, if A is oriented, we loose some

information since what we get are essentially the previous operations applied to thesuspension of X . Note also that the same method may be applied to KA.X/, when A

is modelled on M2.K/.


Another way to proceed is to use a variation of the method developed in [19], p. 21,for any A. We consider the following diagram (with X connected):

F.X; S1/ ��

��

F.X; S1/

��

�nu ��

�

��

U 0.Ayn/ v ��

��

U 0.A y A/yn

��

An �� Aut0.A yn/ �� Aut0.A y A/yn.

In this diagram An is the alternating group (for n � 3), �n is the cartesian product ofAn and U 0.A yn/ over Aut0.A yn/ and the horizontal maps between the U ’s and theAut’s are essentially given by g 7! g ˝ g. Note that these maps induce a map fromF.X; S1/ to itself which is f .z/ 7! f .z2/.

By the general theory developed in 7.2, we know that there exists a canonicalmap � from An to U 0.A y A/

yn such that its composite with the projection fromU 0.A yA/

yn to Aut0.A yA/yn is the composite of the last horizontal maps. Let us

now define the subgroupCn of�n which elementsx are defined by'.�.x// D v.u.x//.It is clear that Cn is a double cover of An which is either the product of An by Z=2 orthe Schur group Cn [40] (since H 2.AnIZ=2/ is isomorphic to Z=2 when n > 3).

Using the methods of §6, we have therefore the following composite maps:

KA.X/ �� KAyn

An.X/ �� KA

yn

.Cn/.X/ ��

KAyn.X/˝R.Cn/,

where the notationKAyn

.Cn/.X/means that the groupCn acts trivially on A

yn. Followingagain Atiyah [2], we see then that any group homomorphism R.Cn/! Z gives rise to

an operation KA.X/! KAyn.X/ Therefore, in the graded case, the Schur group Cn

replaces the symmetric group Sn, which we have used in the ungraded case.In order to define more computable operations, we may replace the alternating

group An by the cyclic group Z=n (when n is odd) as it was already done in [19].The reduction of the central extension Cn becomes trivial and we have a commutativediagram

U 0.Ayn/

��

Z=n

�� Aut.A yn/.

The Adams operation ‰n is then given by the same formula as in 7.4 and [19]

‰n W KA.X/! KA˝n

.X/Zn

146 M. Karoubi

where n is the ring of n-cyclotomic integers. We can show that this operation isadditive and multiplicative up to canonical isomorphisms (see Theorem 30, p. 23 in[19]).

We should also notice, following [9], that we can define the Adams operation ‰�1in graded twisted K-theory as well and combine it with the ‰n’s in order to define

“internal” operations from KA.X/ to KAyn.X/

Zn.

The simplest non-trivial example is

‰n W Z Š K1.S1/! Kn.S1/Zn Š K1.S1/

Zn

where n is a product of different odd primes. Since the operation‰n onK2.S2/ is themultiplication by n, we deduce that � D

p.�1/.n�1/=2n belongs ton (a well-known

result due to Gauss) and that ‰n on K1.S1/ is essentially the inclusion of Z in ndefined by 1 7! � .

As a concluding remark, we should notice that the image of ‰n as defined in 7.8 isnot arbitrary. If k and n are coprimes, the multiplication by k on the group Z=n definesan element of the symmetric group Sn. The signature of this permutation is called theLegendre symbol .kn/. Moreover, this permutation conjugates the elements r and rk inthe group Z=n. If the Legendre symbol is 1, this permutation can be lifted to the Schurgroup Cn. Let us denote now by Fr (as in 7.4) the element of the twisted K-groupassociated to the eigenvalue e2i�r . Then we see that Fr and Frk are isomorphic ifthe Legendre symbol .kn/ is equal to 1 since r and rk are conjugate by an element ofthe Schur group. If n is prime for instance, ‰n.E/ may therefore be written in thefollowing way:

‰n.E/ D F0 CX.kn/D1

U!k CX

.kn/D�1V!k;

where U (resp. V ) is any Fk with Legendre’s symbol equal to 1 (resp. �1). Thisshows in particular that the element � D

p.�1/.n�1/=2n in 7.9 is a “Gauss sum”, a

well-known result.

8 Appendix: A short historical survey of twisted K -theory

Topological K-theory was of course invented by Atiyah and Hirzebruch in 1961 [4]after the fundamental work of Grothendieck [14] and Bott [15]. However, twistedK-theory which is an elaboration of it took some time to emerge. One should quotefirst the work of Atiyah, Bott and Shapiro [6] where Clifford modules and Cliffordbundles (in relation with the Dirac operator) where used to reinterpret Bott periodicityand Thom isomorphism in the presence of Spin or Spinc structures. We then started tounderstand in [28], as quoted in the introduction, thatK-theory of the Thom space maybe defined almost algebraically asK-theory of a functor associated to Clifford bundles.Finally, it was realized in [19] that we can go a step further and consider general gradedalgebra bundles instead of Clifford bundles associated to vector bundles (with a suitablemetric).


On the other hand, the theorem of Atiyah and Jänich [2], [27] showed that Fredholmoperators play a crucial role in K-theory. It was discovered independently in [10] and[29], [32] that Clifford modules may also be used as a variation for the spectrum of(real and complex) K-theory. In particular, the cup-product from Kn.X/ �Kp.Y / toKnCp.X � Y / may be reinterpreted in a much simpler way with this variation. Whatappeared as a convenience for usual K-theory in these previous references became anecessity for the definition of the product in the twisted case, as it was shown in [19].

The next step was taken by J. Rosenberg [38], some twenty years later, in redefiningGBr.X/ for any class in H 3.X IZ/ (not just torsion classes), thanks to the new roleplayed by C*-algebras in K-theory. The need for such a generalization was not clearin the 60’s (although all the tools were there) since we were just interested in usualPoincaré pairing. In passing, we should mention that one can define more generaltwistings in any cohomology theory and therefore in K-theory (see for instance [8],p. 18). However, the geometric ones are the most interesting for the purpose of ourpaper.

We already mentioned the paper of E. Witten [43] in the introduction which madeuse of this more general twisting of J. Rosenberg, together with [35], [8], [9]. Butwe should also quote many other papers: [13] for the relations with the theory ofgerbes and D-branes in Physics; [1] for the relation with orbifolds; [36] who used theChern character defined byA. Connes and M. Karoubi in order to prove an isomorphismbetween twistedK-theory and a computable “twisted cohomology” (at least rationally);[22], [21] about the relation between loop groups and twistedK-theory; [23] (again) forthe relation with equivariant cohomology and many other works which are mentionedinside the paper and in the references. In particular, M. F. Atiyah and M. Hopkins [7]introduced another type of K-theory, denoted by them K˙X/, linked to deep physicalproblems, which is a particular case of twisted equivariant K-theory. This definitionof K˙.X/ was already given in [28], §3.3, in terms of Clifford bundles with a groupaction (see 6.16 and also [31] for the details).

We invite the interested reader to use the classical research tools on the Web formany other references in mathematics and physics.

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[4] M. F. Atiyah and F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull.Amer. Math. Soc. 65 (1959), 276–281.

[5] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, in Proc. Sympos.Pure Math. 3, Amer. Math. Soc., Providence, R.I., 1961, 7–38.

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[8] M. F. Atiyah and G. Segal, Twisted K-theory. Ukr. Mat. Visn. 1 (2004), 287–330; Englishtransl. Ukrain. Math. Bull. 1 (2004), 291–334 .

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[10] M. F. Atiyah and I. M. Singer, Index theory for skew adjoint Fredholm operators, Inst.Hautes Études Sci. Publ. Math. 37 (1969), 5–26 (compare with [32]).

[11] P. Baum andA. Connes, Chern character for discrete groups, in A fête of Topology,AcademicPress, Boston, MA, 1988, 163–232.

[12] B. Blackadar,K-theory for operator algebras, 2nd ed., Math. Sci. Res. Inst. Publ. 5, Cam-bridge University Press, Cambridge 1998.

[13] P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray and D. Stevenson, Twisted K-theoryand K-theory of Bundle Gerbes, Commun. Math. Phys. 228 (2002), 17–45.

[14] A. Borel et J.-P. Serre, Le théorème de Riemann-Roch (d’après Grothendieck), Bull. Soc.Math. France 86 (1958), 97–136.

[15] R. Bott, The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313–337.

[16] A. L. Carey and Bai-Ling Wang, Thom isomorphism and push-forward maps in twistedK-theory, Preprint 2006, math. KT/0507414.

[17] D. Crocker, A. Kumjian, I. Raeburn, D. P. Williams, An equivariant Brauer group and actionof groups on C*-algebras, J. Funct. Anal. 146 (1997), 151–184.

[18] J. Dixmier et A. Douady, Champs continus d’espaces hilbertiens et de C*-algèbres, Bull.Soc. Math. France 91 (1963), 227–284.

[19] P. Donovan and M. Karoubi, Graded Brauer groups and K-theory with local coefficients,Inst. Hautes Études Sci. Publ. Math. 38 (1970), 5–25; French summary in: Groupe de Braueret coefficients locaux enK-théorie, Comptes Rendus Acad. Sci. Paris 269 (1969), 387–389.

[20] A. T. Fomenko and A. S. Miscenko, The index of elliptic operators over C*-algebras, Izv.Akad. Nauk. SSSR Ser. Mat. 43 (1979), 831–859.

[21] D. S. Freed, Twisted K-theory and loop groups, Preprint 2002, arXiv:math/0206237.

[22] D. S. Freed, M. Hopkins and C. Teleman, TwistedK-theory and loop group representations,Preprint 2005, arXiv:math/0312155; Loop Groups and TwistedK-Theory II, Preprint 2007,arXiv:math/0511232.

[23] D. S. Freed, M. Hopkins and C. Teleman, Twisted equivariant K-theory with complexcoefficients, J. Topol. 1 (2008), 16–44.

[24] A. Fröhlich and C. T. C. Wall, Equivariant Brauer groups, in Quadratic forms and theirapplications (Dublin, 1999), Contemp. Math. 272, Amer. Math. Soc., Providence, RI, 2000,57–71.

[25] A. Grothendieck, Le groupe de Brauer, Séminaire Bourbaki Vol. 9, Exp. No. 290, Soc.Math. France, Paris 1995, 199–219.


[26] P. Hu and I. Kriz, TheRO.G/-graded coefficients of .Z=2/n-equivariantK-theory, Preprint2006, ArXiv:math.KT/0609067.

[27] K. Jänich, Vektorraumbündel und der Raum der Fredholm-operatoren. Math. Ann. 161(1965), 129–142.

[28] M. Karoubi, Algèbres de Clifford et K-théorie, Ann. Sci. Ecole Norm. Sup. (4) 1 (1968),161–270.

[29] M. Karoubi, Algèbres de Clifford et opérateurs de Fredholm, in Séminaire Heidelberg-Saarbrücken-Strasbourg sur la K-théorie Lecture Notes in Math. 136, Springer-Verlag,Berlin 1970, 66–106; Summary in Comptes Rendus Acad. Sci. Paris 267 (1968), 305–308.

[30] M. Karoubi, Sur la K-théorie équivariante, in Séminaire Heidelberg-Saarbrücken-Stras-bourg sur la K-théorie Lecture Notes in Math. 136, Springer-Verlag, Berlin 1970, 187–253.

[31] M. Karoubi, EquivariantK-theory of real vector spaces and real projective spaces, TopologyAppl. 122 (2002), 531–546.

[32] M. Karoubi, Espaces classifiants enK-théorie, Trans. Amer. Math. Soc. 147 (1970), 75–115(compare with [10]); Summary in Comptes Rendus Acad. Sci. Paris 268 (1969), 710–713.

[33] N. Kuhn, Character rings in algebraic topology, in Advances in Homotopy Theory (Cortona1988), London Math. Soc. Lecture Note Ser. 139, Cambridge University Press, Cambridge1989, 111–126.

[34] N. Kuiper, The homotopy type of the unitary group in Hilbert space, Topology 3 (1965),19–30.

[35] C. Laurent, J.-L. Tu and P. Xu, Twisted K-theory of differential stacks, Ann. Sci. ÉcoleNorm. Sup. 37 (2004), 841–910.

[36] V. Mathai and D. Stevenson, On a generalized Connes-Hochschild-Kostant-Rosenberg the-orem, Adv. Math. 200 (2006), 303–335.

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[38] J. Rosenberg, Continuous-trace algebras from the bundle theoretic point of view, J. Austral.Math. Soc. A 47 (1989), 368–381.

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[40] I. Schur, Darstellungen der symmetrischen und alternierenden Gruppe durch gebroche lin-eare Substitutionen, J. Reine Angew. Math. 139 (1911), 155–250.

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[42] J.-L. Tu and Ping Xu, The Chern character for twisted K-theory of orbifolds, Adv. Math.207 (2006), 455–483.

[43] E. Witten, D-branes and K-theory, J. High Energy Phys. 12 (1998), paper 19.

[44] R. Wood, Banach algebras and Bott periodicity, Topology 4 (1966), 371–389.

Equivariant cyclic homology for quantum groups

Christian Voigt

1 Introduction

Equivariant cyclic homology can be viewed as a noncommutative generalization ofequivariant de Rham cohomology. For actions of finite groups or compact Lie groups,different aspects of the theory have been studied by various authors [4], [5], [6], [17],[18]. In order to treat noncompact groups as well, a general framework for equivariantcyclic homology following the Cuntz–Quillen formalism [8], [9], [10] has been intro-duced in [25]. For instance, in the setting of discrete groups or totally disconnectedgroups this yields a new approach to classical constructions in algebraic topology [26].However, in contrast to the previous work mentioned above, a crucial feature of theconstruction in [25] is the fact that the basic ingredient in the theory is not a com-plex in the usual sense of homological algebra. In particular, the theory does not fitinto the traditional scheme of defining cyclic homology using cyclic modules or mixedcomplexes.

In this note we define equivariant periodic cyclic homology for quantum groups.This generalizes the constructions in the group case developed in [25]. Again we workin the setting of bornological vector spaces. Correspondingly, the appropriate notionof a quantum group in this context is the concept of a bornological quantum groupintroduced in [27]. This class of quantum groups includes all locally compact groupsand their duals as well as all algebraic quantum groups in the sense of van Daele [24].As in the theory of van Daele, an important ingredient in the definition of a bornologicalquantum group is the Haar measure. It is crucial for the duality theory and also explicitlyused at several points in the construction of the homology theory presented in this paper.However, with some modifications our definition of equivariant cyclic homology couldalso be adapted to a completely algebraic setting using Hopf algebras with invertibleantipodes instead.

From a conceptual point of view, equivariant cyclic homology should be viewedas a homological analogon to equivariant KK-theory [15], [16]. The latter has beenextended by Baaj and Skandalis to coactions of Hopf-C �-algebras [3]. However, inour situation it is more convenient to work with actions instead of coactions.

An important ingredient in equivariant cyclic homology is the concept of a covariantmodule [25]. In the present paper we will follow the terminology introduced in [12]and call these objects anti-Yetter–Drinfeld modules instead. In order to construct thenatural symmetry operator on these modules in the general quantum group setting weprove a formula relating the fourth power of the antipode with the modular functions ofa bornological quantum group and its dual. In the context of finite dimensional Hopfalgebras this formula is a classical result due to Radford [23].

152 C. Voigt

Although anti-Yetter–Drinfeld modules occur naturally in the constructions oneshould point out that our theory does not fit into the framework of Hopf-cyclic co-homology [13]. Still, there are relations to previous constructions for Hopf algebrasby Akbarpour and Khalkhali [1], [2] as well as Neshveyev and Tuset [22]. Remarkin particular that cosemisimple Hopf algebras or finite dimensional Hopf algebras canbe viewed as bornological quantum groups. However, basic examples show that thehomology groups defined in [1], [2], [22] only reflect a small part of the informationcontained in the theory described below.

Let us now describe how the paper is organized. In Section 2 we recall the defini-tion of a bornological quantum group. We explain some basic features of the theoryincluding the definition of the dual quantum group and the Pontrjagin duality theo-rem. This is continued in Section 3 where we discuss essential modules and comodulesover bornological quantum groups as well as actions on algebras and their associatedcrossed products. We prove an analogue of the Takesaki–Takai duality theorem in thissetting. Section 4 contains the discussion of Radford’s formula relating the antipodewith the modular functions of a quantum group and its dual. In Section 5 we studyanti-Yetter–Drinfeld modules over bornological quantum groups and introduce the no-tion of a paracomplex. Section 6 contains a discussion of equivariant differential formsin the quantum group setting. After these preparations we define equivariant periodiccyclic homology in Section 7. In Section 8 we show that our theory is homotopy invari-ant, stable and satisfies excision in both variables. Finally, Section 9 contains a briefcomparison of our theory with the previous approaches mentioned above.

Throughout the paper we work over the complex numbers. For simplicity we haveavoided the use of pro-categories in connection with the Cuntz–Quillen formalism to alarge extent.

2 Bornological quantum groups

The notion of a bornological quantum group was introduced in [27]. We will workwith this concept in our approach to equivariant cyclic homology. For information onbornological vector spaces and more details we refer to [14], [20], [27]. All bornologicalvector spaces are assumed to be convex and complete.

A bornological algebra H is called essential if the multiplication map induces anisomorphismH yH H Š H . The multiplier algebraM.H/ of a bornological algebraH consists of all two-sided multipliers ofH , the latter being defined by the usual alge-braic conditions. There exists a canonical bounded homomorphism � W H ! M.H/.A bounded linear functional � W H ! C on a bornological algebra is called faithfulif �.xy/ D 0 for all y 2 H implies x D 0 and �.xy/ D 0 for all x implies y D 0.If there exists such a functional the map � W H ! M.H/ is injective. In this case onemay view H as a subset of the multiplier algebra M.H/.

In the sequelH will be an essential bornological algebra with a faithful bounded lin-ear functional. For technical reasons we assume moreover that the underlying bornolog-ical vector space of H satisfies the approximation property.

Equivariant cyclic homology for quantum groups 153

A moduleM overH is called essential if the module action induces an isomorphismH yH M Š M . Moreover an algebra homomorphism f W H ! M.K/ is essentialif f turns K into an essential left and right module over H . Assume that � W H !M.H y H/ is an essential homomorphism. The map � is called a comultiplicationif it is coassociative, that is, if .� y id/� D .id y �/� holds. Moreover the Galoismaps �l ; �r ; �l ; �r W H y H !M.H y H/ for � are defined by

�l.x ˝ y/ D �.x/.y ˝ 1/; �r.x ˝ y/ D �.x/.1˝ y/;�l.x ˝ y/ D .x ˝ 1/�.y/; �r.x ˝ y/ D .1˝ x/�.y/:

Let � W H ! M.H y H/ be a comultiplication such that all Galois maps associatedto � define bounded linear maps from H y H into itself. If ! is a bounded linearfunctional on H we define for every x 2 H a multiplier .id y !/�.x/ 2M.H/ by

.id y !/�.x/ � y D .id y !/�l.x ˝ y/;y � .id y !/�.x/ D .id y !/�l.y ˝ x/:

In a similar way one defines .! y id/�.x/ 2 M.H/. A bounded linear functional� W H ! C is called left invariant if

.id y �/�.x/ D �.x/1

for all x 2 H . Analogously one defines right invariant functionals.Let us now recall the definition of a bornological quantum group.

Definition 2.1. A bornological quantum group is an essential bornological algebra Hsatisfying the approximation property with a comultiplication � W H ! M.H y H/such that all Galois maps associated to� are isomorphisms together with a faithful leftinvariant functional � W H ! C.

The definition of a bornological quantum group is equivalent to the definition of analgebraic quantum group in the sense of van Daele [24] in the case that the underlyingbornological vector space carries the fine bornology. The functional � is unique up toa scalar and referred to as the left Haar functional of H .

Theorem 2.2. LetH be a bornological quantum group. Then there exists an essentialalgebra homomorphism � W H ! C and a linear isomorphism S W H ! H which isboth an algebra antihomomorphism and a coalgebra antihomomorphism such that

.� y id/� D id D .id y �/�

and�.S y id/�r D � y id; �.id y S/�l D id y �:

Moreover the maps � and S are uniquely determined.

154 C. Voigt

Using the antipodeS one obtains that every bornological quantum group is equippedwith a faithful right invariant functional as well. Again, such a functional is uniquelydetermined up to a scalar. There are injective bounded linear maps Fl ;Fr ;Gl ;Gr W H !H 0 D Hom.H;C/ defined by the formulas

Fl.x/.h/ D �.hx/; Fr.x/.h/ D �.xh/;Gl.x/.h/ D .hx/; Gr.x/.h/ D .xh/:

The images of these maps coincide and determine a vector space yH . Moreover, thereexists a unique bornology on yH such that these maps are bornological isomorphisms.The bornological vector space yH is equipped with a multiplication which is inducedfrom the comultiplication of H . In this way yH becomes an essential bornologicalalgebra and the multiplication of H determines a comultiplication on yH .

Theorem 2.3. Let H be a bornological quantum group. Then yH with the structuremaps described above is again a bornological quantum group. The dual quantum groupof yH is canonically isomorphic to H .

Explicitly, the duality isomorphism P W H ! yyH is given by P D yGlFlS orequivalently P D yFrGrS . Here we write yGl and yFr for the maps defined aboveassociated to the dual Haar functionals on yH . The second statement of the previoustheorem should be viewed as an analogue of the Pontrjagin duality theorem.

In [27] all calculations were written down explicitly in terms of the Galois mapsand their inverses. However, in this way many arguments tend to become lengthy andnot particularly transparent. To avoid this we shall use the Sweedler notation in thesequel. That is, we write

�.x/ D x.1/ ˝ x.2/for the coproduct of an element x, and accordingly for higher coproducts. Of coursethis has to be handled with care since expressions like the previous one only have aformal meaning. Firstly, the element�.x/ is a multiplier and not contained in an actualtensor product. Secondly, we work with completed tensor products which means thateven a generic element in H y H cannot be written as a finite sum of elementarytensors as in the algebraic case.

3 Actions, coactions and crossed products

In this section we review the definition of essential comodules over a bornologicalquantum group and their relation to essential modules over the dual. Moreover weconsider actions on algebras and their associated crossed products and prove an analogueof the Takesaki–Takai duality theorem.

Let H be a bornological quantum group. Recall from Section 2 that a module VoverH is called essential if the module action induces an isomorphismH yH V Š V .A bounded linear map f W V ! W between essentialH -modules is calledH -linear or


H -equivariant if it commutes with the module actions. We denote the category of es-sentialH -modules and equivariant linear maps byH -Mod. Using the comultiplicationofH one obtains a naturalH -module structure on the tensor product of twoH -modulesand H -Mod becomes a monoidal category in this way.

We will frequently use the regular actions associated to a bornological quantumgroup H . For t 2 H and f 2 yH one defines

t * f D f.1/ f.2/.t/; f ( t D f.1/.t/f.2/and this yields essential left and right H -module structures on yH , respectively.

Dually to the concept of an essential module one has the notion of an essentialcomodule. LetH be a bornological quantum group and let V be a bornological vectorspace. A coaction of H on V is a right H -linear bornological isomorphism W V yH ! V y H such that the relation

.id˝�r/12.id˝��1r / D 1213holds.

Definition 3.1. Let H be a bornological quantum group. An essential H -comodule isa bornological vector space V together with a coaction W V y H ! V y H .

A bounded linear map f W V ! W between essential comodules is called H -colinear if it is compatible with the coactions in the obvious sense. We write Comod-Hfor the category of essential comodules over H with H -colinear maps as morphisms.The category Comod-H is a monoidal category as well.

If the quantum group H is unital, a coaction is the same thing as a bounded linearmap W V ! V y H such that . y id/ D .id y �/ and .id y �/ D id.

Modules and comodules over bornological quantum groups are related in the sameway as modules and comodules over finite dimensional Hopf algebras.

Theorem 3.2. LetH be a bornological quantum group. Every essential leftH -moduleis an essential right yH -comodule in a natural way and vice versa. This yields inverseisomorphisms between the category of essentialH -modules and the category of essen-tial yH -comodules. These isomorphisms are compatible with tensor products.

Since it is more convenient to work with essential modules instead of comoduleswe will usually prefer to consider modules in the sequel.

An essentialH -module is called projective if it has the lifting property with respectto surjections of essentialH -modules with bounded linear splitting. It is shown in [27]that a bornological quantum group H is projective as a left module over itself. Thiscan be generalized as follows.

Lemma 3.3. Let H be a bornological quantum group and let V be any essentialH -module. Then the essential H -modules H y V and V y H are projective.

156 C. Voigt

Proof. Let V� be the space V equipped with the trivialH -action induced by the counit.We have a natural H -linear isomorphism ˛l W H y V ! H y V� given by ˛l.x ˝v/ D x.1/ ˝ S.x.2// � v. Similarly, the map ˛r W V y H ! V� y H given by˛r.v ˝ x/ D S�1.x.1// � v ˝ x.2/ is an H -linear isomorphism. Since H is projectivethis yields the claim.

Using category language an H -algebra is by definition an algebra in the categoryH -Mod. We formulate this more explicitly in the following definition.

Definition 3.4. LetH be a bornological quantum group. AnH -algebra is a bornolog-ical algebra A which is at the same time an essentialH -module such that the multipli-cation map A y A! A is H -linear.

If A is an H -algebra we will also speak of an action of H on A. Remark thatwe do not assume that an algebra has an identity element. The unitarization AC ofan H -algebra A becomes an H -algebra by considering the trivial action on the extracopy C.

According to Theorem 3.2 we can equivalently describe anH -algebra as a bornolog-ical algebra A which is at the same time an essential yH -comodule such that the multi-plication is yH -colinear.

Under additional assumptions there is another possibility to describe this structurewhich resembles the definition of a coaction in the setting of C �-algebras. Let us callan essential bornological algebra A regular if it is equipped with a faithful boundedlinear functional and satisfies the approximation property. If A is regular it followsfrom [27] that the natural bounded linear map A y H !M.A y H/ is injective.

Definition 3.5. Let H be a bornological quantum group. An algebra coaction of Hon a regular bornological algebra A is an essential algebra homomorphism ˛ W A !M.A y H/ such that the coassociativity condition

.˛ y id/˛ D .id y �/˛holds and the maps ˛l and ˛r from A y H to M.A y H/ given by

˛l.a˝ x/ D .1˝ x/˛.a/; ˛r.a˝ x/ D ˛.a/.1˝ x/induce bornological automorphisms of A y H .

It can be shown that an algebra coaction˛ W A!M.A y H/ on a regular bornolog-ical algebra A satisfies .id y �/˛ D id. In particular, the map ˛ is always injective.

Proposition 3.6. Let H be a bornological quantum group and let A be a regularbornological algebra. Then every algebra coaction of yH onA corresponds to a uniqueH -algebra structure on A and vice versa.

Proof. Assume that ˛ is an algebra coaction of yH on A and define D ˛r . Bydefinition is a right yH -linear automorphism ofA y yH . Moreover we have for a 2 A


and f; g 2 yH the relation

.id y �r/12.id y ��1r /.a˝ f ˝ g/ D .id y �r/..˛.a/˝ 1/.1˝ ��1r .f ˝ g///D .id y �/.˛.a//.1˝ �r��1r .f ˝ g//D .˛ y id/.˛.a//.1˝ f ˝ g/D 1213.a˝ f ˝ g/

in M.A y yH y yH/. Using that A is regular we deduce that

.id y �r/12.id y ��1r /.a˝ f ˝ g/ D 1213.a˝ f ˝ g/in A y yH y yH and hence defines a right yH -comodule structure on A. In additionwe have

.� y id/1323.a˝ b ˝ f / D .� y id/13.a˝ ˛.b/.1˝ f //D ˛.a/˛.b/.1˝ f / D ˛.ab/.1˝ f / D .� y id/.a˝ b ˝ f /

and it follows that A becomes an H -algebra using this coaction.Conversely, assume that A is an H -algebra implemented by the coaction

W A y yH ! A y yH . Define bornological automorphisms l and r of A y yH by

l D .id y S�1/�1.id y S/; r D :The map l is left yH -linear for the action of yH on the second tensor factor and r isright yH -linear. Since is a compatible with the multiplication we have

r.� y id/ D .� y id/13r 23r

and

l.� y id/ D .� y id/23l 13l :

In addition one has the equation

.id y �/12l D .id y �/13rrelating l and r . These properties of the maps l and r imply that

˛.a/.b ˝ f / D r.a˝ f /.b ˝ 1/; .b ˝ f /˛.a/ D .b ˝ 1/l.a˝ f /defines an algebra homomorphism ˛ from A toM.A y yH/. As in the proof of Propo-sition 7.3 in [27] one shows that ˛ is essential. Observe that we may identify the naturalmap A yA .A y yH/! A y yH induced by ˛ with 13r W A yA .A y yH y yH yH/!.A yA A/ y . yH y yH yH/ since A is essential.

The maps ˛l and ˛r associated to the homomorphism ˛ can be identified with land r , respectively. Finally, the coaction identity .id y �r/12.id y ��1r / D 1213implies .˛ y id/˛ D .id y �/˛. Hence ˛ defines an algebra coaction of yH on A.

It follows immediately from the constructions that the two procedures describedabove are inverse to each other.

158 C. Voigt

To every H -algebra A one may form the associated crossed product A ÌH . Theunderlying bornological vector space of A Ì H is A y H and the multiplication isdefined by the chain of maps

A y H y A y H �24r �� A y H y A y H id y� y id�� A y A y H � y id

�� A y Hwhere denotes the action ofH on A. Explicitly, the multiplication in AÌH is givenby the formula

.a Ì x/.b Ì y/ D ax.1/ � b ˝ x.2/yfor a; b 2 A and x; y 2 H . On the crossed product A ÌH one has the dual action ofyH defined by

f � .a Ì x/ D a Ì .f * x/

for all f 2 yH . In this wayAÌH becomes an yH -algebra. Consequently one may formthe double crossed productAÌH Ì yH . In the remaining part of this section we discussthe Takesaki–Takai duality isomorphism which clarifies the structure of this algebra.

First we describe a general construction which will also be needed later in connectionwith stability of equivariant cyclic homology. Assume that V is an essentialH -moduleand that A is an H -algebra. Moreover let b W V � V ! C be an equivariant boundedlinear map. We define an H -algebra l.bIA/ by equipping the space V y A y V withthe multiplication

.v1 ˝ a1 ˝ w1/.v2 ˝ a2 ˝ w2/ D b.w1; v2/ v1 ˝ a1a2 ˝ w2and the diagonal H -action.

As a particular case of this construction consider the space V D yH with the regularaction of H given by .t * f /.x/ D f .xt/ and the pairing

ˇ.f; g/ D O .fg/:We write KH for the algebra l.ˇIC/ andA y KH for l.ˇIA/. Remark that the actionon A y KH is not the diagonal action in general. We denote an element f ˝ a˝ g inthis algebra by jf i ˝ a˝ hgj in the sequel. Using the isomorphism yFrS�1 W yH ! H

we identify the above pairing with a pairing H �H ! C. The corresponding actionof H on itself is given by left multiplication and using the normalization � D S. /

we obtain the formula

ˇ.x; y/ D ˇ.SGrS.x/; SGrS.y// D ˇ.Fl.x/;Fl.y// D �.S�1.y/x/ D .S.x/y/for the above pairing expressed in terms of H .

Let H be a bornological quantum group and let A be an H -algebra. We define abounded linear map �A W A ÌH Ì yH ! A y KH by

�A.a Ì x Ì Fl.y// D jy.1/S.x.2//i ˝ y.2/S.x.1// � a˝ hy.3/jand it is easily verified that �A is an equivariant bornological isomorphism. In addition, astraightforward computation shows that�A is an algebra homomorphism. Consequentlywe obtain the following analogue of the Takesaki–Takai duality theorem.


Proposition 3.7. LetH be a bornological quantum group and let A be anH -algebra.Then the map �A W A ÌH Ì yH ! A y KH is an equivariant algebra isomorphism.

For algebraic quantum groups a discussion of Takesaki–Takai duality is containedin [11]. More information on similar duality results in the context of Hopf algebras canbe found in [21].

IfH D D.G/ is the smooth convolution algebra of a locally compact groupG thenanH -algebra is the same thing as aG-algebra. As a special case of Proposition 3.7 oneobtains that for everyG-algebraA the double crossed productAÌH Ì yH is isomorphicto the G-algebra A y KG used in [25].

4 Radford’s formula

In this section we prove a formula for the fourth power of the antipode in terms of themodular elements of a bornological quantum group and its dual. This formula wasobtained by Radford in the setting of finite dimensional Hopf algebras [23].

Let H be a bornological quantum group. If � is a left Haar functional on H thereexists a unique multiplier ı 2M.H/ such that

.� y id/�.x/ D �.x/ıfor all x 2 H . The multiplier ı is called the modular element of H and measuresthe failure of � from being right invariant. It is shown in [27] that ı is invertible withinverse S.ı/ D S�1.ı/ D ı�1 and that one has �.ı/ D ı ˝ ı as well as �.ı/ D 1.In terms of the dual quantum group the modular element ı defines a character, that is,an essential homomorphism from yH to C. Similarly, there exists a unique modularelement Oı 2M. yH/ for the dual quantum group which satisfies

. O� y id/ O�.f / D O�.f / Oıfor all f 2 yH .

The Haar functionals of a bornological quantum group are uniquely determined upto a scalar multiple. In many situations it is convenient to fix a normalization at somepoint. However, in the discussion below it is not necessary to keep track of the scalingof the Haar functionals. If ! and are linear functionals we shall write ! � if thereexists a nonzero scalar such that ! D . We use the same notation for elements in abornological quantum group or linear maps that differ by some nonzero scalar multiple.Moreover we shall identify H with its double dual using Pontrjagin duality.

To begin with observe that the bounded linear functional ı * � on H defined by

.ı * �/.x/ D �.xı/is faithful and satisfies

..ı * �/ y id/�.x/ D .� y id/.�.xı/.1˝ ı�1// D �.xı/ıı�1 D .ı * �/.x/:

160 C. Voigt

It follows that ı * � is a right Haar functional on H . In a similar way we obtain aright Haar functional � ( ı on H . Hence

ı * � � � � ( ı

by uniqueness of the right Haar functional which yields in particular the relations

Fl.xı/ � Gl.x/; Fr.ıx/ � Gr.x/

for the Fourier transform.According to Pontrjagin duality we have x D yGlFlS.x/ for all x 2 H . Since

yGl.f / � yFl.f Oı/ for every f 2 yH as well as

.Fl.S.x// Oı/.h/ D �.h.1/S.x// Oı.h.2// D �.hS.x.2/// Oı.ıS2.x.1///� Fl.S.x.2///.h/ Oı.x.1// D Fl.S.x ( Oı//.h/

we obtain x � yFlFlS.x ( Oı/ or equivalently

S�1. Oı * x/ � yFlFl.x/: (4.1)

Using equation (4.1) and the formula x � S�1 yFlFr.x/ obtained from Pontrjaginduality we compute

yFlFl. Oı�1 * S2.x// � S�1S2.x/ D S.x/ � yFlFr.x/which implies

Fl.S2.x// � Fr. Oı * x/ (4.2)

since yFl is an isomorphism. Similarly, we have yFlSGl � id and using yFl.f / �yGl.f Oı�1/ together with

.SGl.x/ Oı�1/.h/ D .S.h.1//x/ Oı�1.h.2//� .S.h/x.2// Oı�1.x.1// D SGl.x ( Oı�1/.h/

we obtain yGlSGl.x/ � x ( Oı. According to the relation Fl.xı/ � Gl.x/ this may berewritten as S�1 yFrFl.xı/ � x ( Oı which in turn yields

yFrFl.x/ � S..xı�1/ ( Oı/: (4.3)

Due to Pontrjagin duality we have x D yFrGrS.x/ for all x 2 H and using Gr.x/ �Fr.ıx/ we obtain

yFrFr.S.x// � yFrGr.ı�1S.x// D xı: (4.4)

According to equation (4.3) and equation (4.4) we have

yFrFl.S�2.ı�1.x ( Oı�1/ı// � S..S�2.ı�1.x ( Oı�1/ı/ı�1/ ( Oı/� S.S�2.ı�1x// D S�1.ı�1x/ � yFrFr.x/


and since yFr is an isomorphism this implies

Fr.S2.x// � Fl.ı

�1.x ( Oı�1/ı/: (4.5)

Assembling these relations we obtain the following result.

Proposition 4.1. LetH be a bornological quantum group and let ı and Oı be the modularelements of H and yH , respectively. Then

S4.x/ D ı�1. Oı * x ( Oı�1/ıfor all x 2 H .

Proof. Using equation (4.2) and equation (4.5) we compute

Fl.S4.x// � Fr. Oı * S2.x// D Fr.S

2. Oı * x// � Fl.ı�1. Oı * x ( Oı�1/ı/

which impliesS4.x/ � ı�1. Oı * x ( Oı�1/ı

for all x 2 H since Fl is an isomorphism. The claim follows from the observation thatboth sides of the previous equation define algebra automorphisms of H .

5 Anti-Yetter–Drinfeld modules

In this section we introduce the notion of an anti-Yetter–Drinfeld module over a borno-logical quantum group. Moreover we discuss the concept of a paracomplex.

We begin with the definition of an anti-Yetter–Drinfeld module. In the context ofHopf algebras this notion was introduced in [12].

Definition 5.1. Let H be a bornological quantum group. An H -anti-Yetter–Drinfeldmodule is an essential leftH -moduleM which is also an essential left yH -module suchthat

t � .f �m/ D .S2.t.1// * f ( S�1.t.3/// � .t.2/ �m/:for all t 2 H;f 2 yH and m 2 M . A homomorphism � W M ! N between anti-Yetter–Drinfeld modules is a bounded linear map which is bothH -linear and yH -linear.

We will not always mention explicitly the underlying bornological quantum groupwhen dealing with anti-Yetter–Drinfeld modules. Moreover we shall use the abbrevia-tions AYD-module and AYD-map for anti-Yetter–Drinfeld modules and their homomor-phisms.

According to Theorem 3.2 a left yH -module structure corresponds to a right H -comodule structure. Hence an AYD-module can be described equivalently as a bornolog-ical vector spaceM equipped with an essentialH -module structure and anH -comodulestructure satisfying a certain compatibility condition. Formally, this compatibility con-dition can be written down as

.t �m/.0/ ˝ .t �m/.1/ D t.2/ �m.0/ ˝ t.3/m.1/S.t.1//for all t 2 H and m 2M .

162 C. Voigt

We want to show that AYD-modules can be interpreted as essential modules overa certain bornological algebra. Following the notation in [12] this algebra will bedenoted by A.H/. As a bornological vector space we have A.H/ D yH y H and themultiplication is defined by the formula

.f ˝ x/ � .g ˝ y/ D f .S2.x.1// * g ( S�1.x.3///˝ x.2/y:There exists an algebra homomorphism �H W H !M.A.H// given by

�H .x/ � .g ˝ y/ D S2.x.1// * g ( S�1.x.3//˝ x.2/yand

.g ˝ y/ � �H .x/ D g ˝ yx:It is easily seen that �H is injective. Similarly, there is an injective algebra homomor-phism � yH W yH !M.A.H// given by

� yH .f / � .g ˝ y/ D fg ˝ yas well as

.g ˝ y/ � � yH .f / D g.S2.y.1// * f ( S�1.y.3///˝ y.2/and we have the following result.

Proposition 5.2. For every bornological quantum group H the bornological algebraA.H/ is essential.

Proof. The homomorphism � yH induces on A.H/ the structure of an essential left yH -module. Similarly, the space A.H/ becomes an essential left H -module using thehomomorphism �H . In fact, if we write yH� for the space yH equipped with the trivialH -action the map c W A.H/! yH� y H given by

c.f ˝ x/ D S.x.1// * f ( x.3/ ˝ x.2/is an H -linear isomorphism. Actually, the actions of yH and H on A.H/ are definedin such a way that A.H/ becomes an AYD-module. Using the canonical isomorphismH yH A.H/ Š A.H/ one obtains an essential yH -module structure on H yH A.H/given explicitly by the formula

f � .x ˝ g ˝ y/ D x.2/ ˝ .S.x.1// * f ( x.3//g ˝ y:

Correspondingly, we obtain a natural isomorphism yH y yH .H yH A.H// Š A.H/.

It is straightforward to verify that the identity map yH y .H y A.H// Š yH y H yyH y H ! A.H/ y A.H/ induces an isomorphism yH y yH .H yH A.H// !

A.H/ y A.H/ A.H/. This yields the claim.

We shall now characterize AYD-modules as essential modules over A.H/.


Proposition 5.3. Let H be a bornological quantum group. Then the category ofAYD-modules over H is isomorphic to the category of essential left A.H/-modules.

Proof. Let M Š A.H/ y A.H/ M be an essential A.H/-module. Then we obtaina left H -module structure and a left yH -module structure on M using the canonicalhomomorphisms �H W H ! M.A.H// and � yH W yH ! M.A.H//. Since the action ofH on A.H/ is essential we have natural isomorphisms

H yH M Š H yH A.H/ y A.H/ M Š A.H/ y A.H/ M ŠMand hence M is an essential H -module. Similarly we have

yH y yH M Š yH y yH A.H/ y A.H/ y M Š A.H/ y A.H/ M ŠM

since A.H/ is an essential yH -module. These module actions yield the structure of anAYD-module on M .

Conversely, assume thatM is anH -AYD-module. Then we obtain an A.H/-modulestructure on M by setting

.f ˝ t / �m D f � .t �m/for f 2 yH and t 2 H . Since M is an essential H -module we have a naturalisomorphism H yH M Š M . As in the proof of Proposition 5.2 we obtain aninduced essential yH -module structure on H yH M and canonical isomorphismsA.H/ y A.H/ M Š yH y yH .H yH M/ Š M . It follows that M is an essentialA.H/-module.

The previous constructions are compatible with morphisms and it is easy to checkthat they are inverse to each other. This yields the assertion.

There is a canonical operator T on every AYD-module which plays a crucial rolein equivariant cyclic homology. In order to define this operator it is convenient topass from yH to H in the first tensor factor of A.H/. More precisely, consider thebornological isomorphism W A.H/! H y H given by

.f ˝ y/ D yFl.f /˝ y ( Oı�1

where Oı 2M. yH/ is the modular function of yH . The inverse map is given by

�1.x ˝ y/ D SGl.x/˝ y ( Oı:It is straightforward to check that the left H -action on A.H/ corresponds to

t � .x ˝ y/ D t.3/xS.t.1//˝ t.2/y

and the left yH -action becomes

f � .x ˝ y/ D .f * x/˝ y

164 C. Voigt

under this isomorphism. The right H -action is identified with

.x ˝ y/ � t D x ˝ y.t ( Oı�1/and the right yH -action corresponds to

.x ˝ y/ � g D x.2/.S2.y.2// * g ( S�1.y.4///.S�2.x.1//ı/˝ Oı.y.1//y.3/ ( Oı�1

where ı is the modular function of H . Using this description of A.H/ we obtain thefollowing result.

Proposition 5.4. The bounded linear map T W A.H/! A.H/ defined by

T .x ˝ y/ D x.2/ ˝ S�1.x.1//yis an isomorphism of A.H/-bimodules.

Proof. It is evident that T is a bornological isomorphism with inverse given byT �1.x ˝ y/ D x.2/ ˝ x.1/y. We compute

T .t � .x ˝ y// D T .t.3/xS.t.1//˝ t.2/y/D t.5/x.2/S.t.1//˝ S�1.t.4/x.1/S.t.2///t.3/yD t.3/x.2/S.t.1//˝ t.2/S�1.x.1//yD t � T .x ˝ y/

and it is clear that T is left yH -linear. Consequently T is a left A.H/-linear map.Similarly, we have

T ..x˝ y/ � t / D T .x˝ y.t ( Oı�1// D x.2/˝S�1.x.1//y.t ( Oı�1/ D T .x˝ y/ � tand hence T is right H -linear. In order to prove that T is right yH -linear we compute

T �1..x ˝ y/ � g/D T �1.x.2/.S2.y.2// * g ( S�1.y.4///.S�2.x.1//ı/˝ Oı.y.1//y.3/ ( Oı�1/D x.3/.S2.y.2// * g ( S�1.y.4///.S�2.x.1//ı/˝ Oı.y.1//x.2/.y.3/ ( Oı�1/D x.3/.S2.y.2// * g ( S�1.y.4///.S�2. Oı * x.1//ı/˝ Oı.y.1//.x.2/y.3// ( Oı�1

which according to Proposition 4.1 is equal to

D x.3/.S2.y.2// * g ( S�1.y.4///.ıS2.x.1/ ( Oı//˝ Oı.y.1//.x.2/y.3// ( Oı�1D x.6/.S2.x.2//S2.y.2// * g ( S�1.y.4//S�1.x.4///.S�2.x.5//ı/˝ Oı.x.1/y.1//.x.3/y.3// ( Oı�1

D .x.2/ ˝ x.1/y/ � gD T �1.x ˝ y/ � g:

We conclude that T is a right A.H/-linear map as well.


If M is an arbitrary AYD-module we define T W M !M by

T .F ˝m/ D T .F /˝mfor F ˝m 2 A.H/ y A.H/ M . Due to Proposition 5.3 and Lemma 5.4 this definitionmakes sense.

Proposition 5.5. The operator T defines a natural isomorphism T W id ! id of theidentity functor on the category of AYD-modules.

Proof. It is clear from the construction that T W M !M is an isomorphism for allM .If � W M ! N is an AYD-map the equation T � D �T follows easily after identifying �with the map id y � W A.H/ y A.H/ M ! A.H/ y A.H/ N . This yields the assertion.

If the bornological quantum group H is unital one may construct the operator Ton an AYD-module M directly from the coaction M !M y H corresponding to theaction of yH . More precisely, one has the formula

T .m/ D S�1.m.1// �m.0/for every m 2M .

Using the terminology of [25] it follows from Proposition 5.5 that the category ofAYD-modules is a para-additive category in a natural way. This leads in particular tothe concept of a paracomplex of AYD-modules.

Definition 5.6. A paracomplex C D C0˚C1 consists of AYD-modules C0 and C1 andAYD-maps @0 W C0 ! C1 and @1 W C1 ! C0 such that

@2 D id�Twhere the differential @ W C ! C1 ˚ C0 Š C is the composition of @0 ˚ @1 with thecanonical flip map.

The morphism @ in a paracomplex is called a differential although it usually does notsatisfy the relation @2 D 0. As for ordinary complexes one defines chain maps betweenparacomplexes and homotopy equivalences. We always assume that such maps arecompatible with the AYD-module structures. Let us point out that it does not makesense to speak about the homology of a paracomplex in general.

The paracomplexes we will work with arise from paramixed complexes in the fol-lowing sense.

Definition 5.7. A paramixed complex M is a sequence of AYD-modules Mn togetherwith AYD-maps b of degree �1 and B of degreeC1 satisfying b2 D 0, B2 D 0 and

Œb; B� D bB C Bb D id�T:If T is equal to the identity operator on M this reduces of course to the definition

of a mixed complex.

166 C. Voigt

6 Equivariant differential forms

In this section we define equivariant differential forms and the equivariantX -complex.Moreover we discuss the properties of the periodic tensor algebra of an H -algebra.These are the main ingredients in the construction of equivariant cyclic homology.

Let H be a bornological quantum group. If A is an H -algebra we obtain a leftaction of H on the space H y n.A/ by

t � .x ˝ !/ D t.3/xS.t.1//˝ t.2/ � !

for t; x 2 H and ! 2 n.A/. Here n.A/ D AC y A yn for n > 0 is the space ofnoncommutative n-forms over A with the diagonal H -action. For n D 0 one defines 0.A/ D A. There is a left action of the dual quantum group yH onH y n.A/ givenby

f � .x ˝ !/ D .f * x/˝ ! D f .x.2//x.1/ ˝ !:By definition, the equivariant n-forms nH .A/ are the spaceH y n.A/ together withthe H -action and the yH -action described above. We compute

t � .f � .x ˝ !// D t � .f .x.2//x.1/ ˝ !/D f .x.2//t.3/x.1/S.t.1//˝ t.2/ � !D .S2.t.1// * f ( S�1.t.5/// � .t.4/xS.t.2//˝ t.3/ � !/D .S2.t.1// * f ( S�1.t.3/// � .t.2/ � .x ˝ !//

and deduce that nH .A/ is anH -AYD-module. We let H .A/ be the direct sum of thespaces nH .A/.

Now we define operators d and bH on H .A/ by

d.x ˝ !/ D x ˝ d!

andbH .x ˝ !da/ D .�1/j!j.x ˝ !a � x.2/ ˝ .S�1.x.1// � a/!/:

The map bH should be thought of as a twisted version of the usual Hochschild operator.We compute

b2H .x ˝ !dadb/ D .�1/j!jC1bH .x ˝ !dab � x.2/ ˝ .S�1.x.1// � b/!da/D .�1/j!jC1bH .x ˝ !d.ab/ � x ˝ !adb � x.2/ ˝ .S�1.x.1// � b/!da/D �.x ˝ !ab � x.2/ ˝ S�1.x.1// � .ab/! � x ˝ !ab C x.2/ ˝ .S�1.x.1// � b/!a� x.2/ ˝ .S�1.x.1// � b/!aC x.2/ ˝ S�1.x.1// � .ab/!/ D 0

which shows that b2H is a differential as in the nonequivariant situation. Let us discussthe compatibility of d and bH with the AYD-module structure. It is easy to check that


d is an AYD-map and that the operator bH is yH -linear. Moreover we compute

bH .t � .x ˝ !da// D .�1/j!j.t.4/xS.t.1//˝ .t.2/ � !/.t.3/ � a/� t.6/x.2/S.t.1//˝ .S�1.t.5/x.1/S.t.2///t.4/ � a/.t.3/ � !//

D .�1/j!j.t.3/xS.t.1//˝ t.2/ � .!a/ � t.4/x.2/S.t.1//˝ .t.2/S�1.x.1// � a/.t.3/ � !//D t � bH .x ˝ !da/

and deduce that bH is an AYD-map as well.Similar to the non-equivariant case we used and bH to define an equivariant Karoubi

operator �H and an equivariant Connes operator BH by

�H D 1 � .bHd C dbH /and

BH DnX

jD0�jHd

on nH .A/. Let us record the following explicit formulas. For n > 0 we have

�H .x ˝ !da/ D .�1/n�1x.2/ ˝ .S�1.x.1// � da/!on nH .A/ and in addition �H .x ˝ a/ D x.2/ ˝ S�1.x.1// � a on 0H .A/. For theConnes operator we compute

BH .x˝a0da1 : : : dan/DnXiD0.�1/nix.2/˝S�1.x.1//�.danC1�i : : : dan/:da0 : : : dan�i

Furthermore, the operator T is given by

T .x ˝ !/ D x.2/ ˝ S�1.x.1// � !on equivariant differential forms. Observe that all operators constructed so far areAYD-maps and thus commute with T according to Proposition 5.5.

Lemma 6.1. On nH .A/ the following relations hold:

a) �nC1H d D Td ,

b) �nH D T C bH�nHd ,

c) �nHbH D bHT ,

d) �nC1H D .id�dbH /T ,

e) .�nC1H � T /.�nH � T / D 0,

f) BHbH C bHBH D id�T .

168 C. Voigt

Proof. a) follows from the explicit formula for �H . b) We compute

�nH .x ˝ a0da1 : : : dan/ D x.2/ ˝ S�1.x.1// � .da1 : : : dan/a0D x.2/ ˝ S�1.x.1// � .a0da1 : : : dan/C .�1/nbH .x.2/ ˝ S�1.x.1// � .da1 : : : dan/da0/

D x.2/ ˝ S�1.x.1// � .a0da1 : : : dan/C bH�nHd.x ˝ a0da1 : : : dan/which yields the claim. c) follows by applying bH to both sides of b). d) Apply �H tob) and use a). e) is a consequence of b) and d). f) We compute

BHbH C bHBH Dn�1XjD0

�jHdbH C

nXjD0

bH�jHd D

n�1XjD0

�jH .dbH C bHd/C �nHbHd

D id��nH .1 � bHd/ D id��nH .�H C dbH /D id�T C dbHT � TdbH D id�T

where we use d) and b).

From the definition of BH and the fact that d2 D 0 we obtain B2H D 0. Let ussummarize this discussion as follows.

Proposition 6.2. LetA be anH -algebra. The space H .A/ of equivariant differentialforms is a paramixed complex in the category of AYD-modules.

We remark that the definition of H .A/ for H D D.G/ differs slightly from thedefinition of G.A/ in [25] if the locally compact groupG is not unimodular. However,this does not affect the definition of the equivariant homology groups.

In the sequel we will drop the subscripts when working with the operators on H .A/introduced above. For instance, we shall simply write b instead of bH and B insteadof BH .

The n-th level of the Hodge tower associated to H .A/ is defined by

�n H .A/ Dn�1MjD0

jH .A/˚ nH .A/=b. nC1H .A//:

Using the grading into even and odd forms we see that �n H .A/ together with theboundary operator B C b becomes a paracomplex. By definition, the Hodge tower� H .A/ of A is the projective system .�n H .A//n2N obtained in this way.

From a conceptual point of view it is convenient to work with pro-categories in thesequel. The pro-category pro.C/ over a category C consists of projective systems inC . A pro-H -algebra is simply an algebra in the category pro.H -Mod/. For instance,everyH -algebra becomes a pro-H -algebra by viewing it as a constant projective system.More information on the use of pro-categories in connection with cyclic homology canbe found in [10], [25].


Definition 6.3. Let A be a pro-H -algebra. The equivariant X -complex XH .A/ of Ais the paracomplex �1 H .A/. Explicitly, we have

XH .A/ W 0H .A/d ��

1H .A/=b. 2H .A//:

b��

We are interested in particular in the equivariant X -complex of the periodic tensoralgebra T A of an H -algebra A. The periodic tensor algebra T A is the even part of� .A/ equipped with the Fedosov product given by

! ı D ! � .�1/j!jd!dfor homogenous forms ! and . The natural projection � .A/ ! A restricts to anequivariant homomorphism �A W T A! A and we obtain an extension

JA �� T A�A �� A

of pro-H -algebras.The main properties of the pro-algebras T A and JA are explained in [20], [25]. Let

us recall some terminology. We write �n W N yn ! N for the iterated multiplicationin a pro-H -algebra N . Then N is called locally nilpotent if for every equivariant pro-linear map f W N ! C with constant range C there exists n 2 N such that f�n D 0.It is straightforward to check that the pro-H -algebra JA is locally nilpotent.

An equivariant pro-linear map l W A ! B between pro-H -algebras is called alonilcur if its curvature !l W A y A ! B defined by !l.a; b/ D l.ab/ � l.a/l.b/is locally nilpotent, that is, if for every equivariant pro-linear map f W B ! C with

constant range C there exists n 2 N such that f�nB!ynlD 0. The term lonilcur is an

abbreviation for "equivariant pro-linear map with locally nilpotent curvature". SinceJA is locally nilpotent the natural map �A W A! T A is a lonilcur.

Proposition 6.4. Let A be an H -algebra. The pro-H -algebra T A and the lonilcur�A W A! T A satisfy the following universal property. If l W A! B is a lonilcur intoa pro-H -algebra B there exists a unique equivariant homomorphism �l� W T A ! B

such that �l��A D l .An important ingredient in the Cuntz–Quillen approach to cyclic homology [8], [9],

[10] is the concept of a quasifree pro-algebra. The same is true in the equivariant theory.

Definition 6.5. A pro-H -algebra R is called H -equivariantly quasifree if there existsan equivariant splitting homomorphism R! T R for the projection �R.

We state some equivalent descriptions of equivariantly quasifree pro-H -algebras.

Theorem 6.6. LetH be a bornological quantum group and letR be a pro-H -algebra.The following conditions are equivalent:

a) R is H -equivariantly quasifree.

170 C. Voigt

b) There exists an equivariant pro-linear map r W 1.R/! 2.R/ satisfying

r.a!/ D ar.!/; r.!a/ D r.!/a � !dafor all a 2 R and ! 2 1.R/.

c) There exists a projective resolution 0! P1 ! P0 ! RC of the R-bimodule RCof length 1 in pro.H -Mod/.

A map r W 1.R/! 2.R/ satisfying condition b) in Theorem 6.6 is also calledan equivariant graded connection on 1.R/.

We have the following basic examples of quasifree pro-H -algebras.

Proposition 6.7. LetA be anyH -algebra. The periodic tensor algebra T A isH -equi-variantly quasifree.

An important result in theory of Cuntz and Quillen relates the X -complex of theperiodic tensor algebra T A to the standard complex of A constructed using noncom-mutative differential forms. The comparison between the equivariant X -complex andequivariant differential forms is carried out in the same way as in the group case [25].

Proposition 6.8. There is a natural isomorphism XH .T A/ Š � H .A/ such that thedifferentials of the equivariant X -complex correspond to

@1 D b � .idC�/d on � oddH .A/

@0 D �n�1XjD0

�2j b C B on 2nH .A/:

Theorem 6.9. Let H be a bornological quantum group and let A be an H -algebra.Then the paracomplexes � H .A/ and XH .T A/ are homotopy equivalent.

For the proof of Theorem 6.9 it suffices to observe that the corresponding argumentsin [25] are based on the relations obtained in Proposition 6.1.

7 Equivariant periodic cyclic homology

In this section we define equivariant periodic cyclic homology for bornological quantumgroups.

Definition 7.1. LetH be a bornological quantum group and letA andB beH -algebras.The equivariant periodic cyclic homology of A and B is

HPH� .A;B/ D H�.HomA.H/.XH .T .A ÌH Ì yH//;XH .T .B ÌH Ì yH///:We write HomA.H/ for the space of AYD-maps and consider the usual differential for

a Hom-complex in this definition. Using Proposition 5.5 it is straightforward to check


that this yields indeed a complex. Remark that both entries in the above Hom-complexare only paracomplexes.

Let us consider the special case that H D D.G/ is the smooth group algebraof a locally compact group G. In this situation the definition of HPH� reduces tothe definition of HPG� given in [25]. This is easily seen using the Takesaki–Takaiisomorphism obtained in Proposition 3.7 and the results from [27].

As in the group case HPH� is a bifunctor, contravariant in the first variable andcovariant in the second variable. We defineHPH� .A/ D HPH� .C; A/ to be the equiv-ariant periodic cyclic homology of A and HP �H .A/ D HPH� .A;C/ to be equivariantperiodic cyclic cohomology. There is a natural associative product

HPHi .A;B/ �HPHj .B; C /! HPHiCj .A; C /; .x; y/ 7! x � yinduced by the composition of maps. Every equivariant homomorphism f W A ! B

defines an element in HPH0 .A;B/ denoted by Œf �. The element Œid� 2 HPH0 .A;A/is denoted 1 or 1A. An element x 2 HPH� .A;B/ is called invertible if there exists anelement y 2 HPH� .B;A/ such that x � y D 1A and y � x D 1B . An invertible elementof degree zero is called anHPH -equivalence. Such an element induces isomorphismsHPH� .A;D/ Š HPH� .B;D/ andHPH� .D;A/ Š HPH� .D;B/ for allH -algebrasD.

8 Homotopy invariance, stability and excision

In this section we show that equivariant periodic cyclic homology is homotopy invariant,stable and satisfies excision in both variables. Since the arguments carry over from thegroup case with minor modifications most of the proofs will only be sketched. Moredetails can be found in [25].

We begin with homotopy invariance. Let B be a pro-H -algebra and consider theFréchet algebra C1Œ0; 1� of smooth functions on the interval Œ0; 1�. We denote byBŒ0; 1� the pro-H -algebra B y C1Œ0; 1� where the action on C1Œ0; 1� is trivial. A(smooth) equivariant homotopy is an equivariant homomorphism ˆ W A ! BŒ0; 1� ofH -algebras. Evaluation at the point t 2 Œ0; 1� yields an equivariant homomorphismˆt W A ! B . Two equivariant homomorphisms from A to B are called equivariantlyhomotopic if they can be connected by an equivariant homotopy.

Theorem 8.1 (Homotopy invariance). Let A and B be H -algebras and let ˆ W A !BŒ0; 1� be a smooth equivariant homotopy. Then the elements Œˆ0� and Œˆ1� inHPH0 .A;B/ are equal. Hence the functor HPH� is homotopy invariant in both vari-ables with respect to smooth equivariant homotopies.

Recall that �2 H .A/ is the paracomplex 0H .A/˚ 1H .A/˚ 2H .A/=b. 3H .A//with the usual differential B C b and the grading into even and odd forms for anypro-H -algebra A. There is a natural chain map �2 W �2 H .A/! XH .A/.

Proposition 8.2. Let A be an equivariantly quasifree pro-H -algebra. Then the map�2 W �2 H .A/! XH .A/ is a homotopy equivalence.

172 C. Voigt

A homotopy inverse is constructed using an equivariant connection for 1.A/.Now let ˆ W A ! BŒ0; 1� be an equivariant homotopy. The derivative of ˆ is an

equivariant linear map ˆ0 W A! BŒ0; 1�. If we view BŒ0; 1� as a bimodule over itselfthe map ˆ0 is a derivation with respect to ˆ in the sense that ˆ0.ab/ D ˆ0.a/ˆ.b/Cˆ.a/ˆ0.b/ for a; b 2 A. We define an AYD-map W nH .A/! n�1H .B/ for n > 0 by

.x ˝ a0da1 : : : dan/ DZ 1

0

x ˝ˆt .a0/ˆ0t .a1/dˆt .a2/ : : : dˆt .an/dt

and an explicit calculation yields the following result.

Lemma 8.3. We have XH .ˆ1/�2 � XH .ˆ0/�2 D @ C @. Hence the chain mapsXH .ˆt /�

2 W �2 H .A/! XH .B/ for t D 0; 1 are homotopic.

Using the mapˆwe obtain an equivariant homotopyA y KH ! .B y KH /Œ0; 1�

which induces an equivariant homomorphism T .A y KH / ! T ..B y KH /Œ0; 1�/.Together with the obvious homomorphism T ..B y KH /Œ0; 1�/! T .B y KH /Œ0; 1�

this yields an equivariant homotopy ‰ W T .A y KH / ! T .B y KH /Œ0; 1�. SinceT .A y KH / is equivariantly quasifree we can apply Proposition 8.2 and Lemma 8.3to obtain Œˆ0� D Œˆ1� 2 HPH0 .A;B/. This finishes the proof of Theorem 8.1.

Homotopy invariance has several important consequences. Let us call an extension0 ! J ! R ! A ! 0 of pro-H -algebras with equivariant pro-linear splitting auniversal locally nilpotent extension ofA if J is locally nilpotent andR is equivariantlyquasifree. In particular, 0 ! JA ! T A ! A ! 0 is a universal locally nilpotentextension of A. Using homotopy invariance one shows that HPH� can be computedusing arbitrary universal locally nilpotent extensions.

Let us next study stability. One has to be slightly careful to formulate correctlythe statement of the stability theorem since the tensor product of twoH -algebras is nolonger an H -algebra in general.

Let H be a bornological quantum group and assume that we are given an essentialH -module V together with an equivariant bilinear pairing b W V � V ! C. Moreoverlet A be an H -algebra. Recall from Section 3 that l.bIA/ D V y A y V is theH -algebra with multiplication

.v1 ˝ a1 ˝ w1/.v2 ˝ a2 ˝ w2/ D b.w1; v2/ v1 ˝ a1a2 ˝ w2and the diagonal H -action. We call the pairing b admissible if there exists an H -invariant vector u 2 V such that b.u; u/ D 1. In this case the map �A W A ! l.bIA/given by �.a/ D u˝ a˝ u is an equivariant homomorphism.

Theorem 8.4. LetH be a bornological quantum group and letA be anH -algebra. Forevery admissible equivariant bilinear pairing b W V �V ! C the map � W A! l.bIA/induces an invertible element Œ�A� 2 H0.HomA.H/.XH .T A/;XH .T .l.bIA////.Proof. The canonical map l.bIA/! l.bI T A/ induces an equivariant homomorphismA W T .l.bIA//! l.bI T A/. Define the map trA W XH .l.bI T A//! XH .T A/ by

trA.x ˝ .v0 ˝ a0 ˝ w0// D b.S�1.x.1// � w0; v0/ x.2/ ˝ a0


and

trA.x ˝ .v0 ˝ a0 ˝ w0/ d.v1 ˝ a1 ˝ w1//D b..S�1.x.1// � w1; v0/b.w0; v1/ x.2/ ˝ a0da1:

In these formulas we implicitly use the twisted trace trx W l.b/! C for x 2 H definedby trx.v ˝ w/ D b..S�1.x/ � w; v/. The twisted trace satisfies the relation

trx.T0T1/ D trx.2/..S�1.x.1// � T1/T0/

for all T0; T1 2 l.b/. Using this relation one verifies that trA defines a chain map. It isclear that trA is yH -linear and it is straightforward to check that trA is H -linear. Let usdefine tA D trA ıXH .A/ and show that ŒtA� is an inverse for Œ�A�. Using the fact that uis H -invariant one computes Œ�A� � ŒtA� D 1. We have to prove ŒtA� � Œ�A� D 1. Considerthe following equivariant homomorphisms l.bIA/! l.bI l.bIA// given by

i1.v ˝ a˝ w/ D u˝ v ˝ a˝ w ˝ u;i2.v ˝ a˝ w/ D v ˝ u˝ a˝ u˝ w:

As above we see Œi1� � Œtl.bIA/� D 1 and we determine Œi2� � Œtl.bIA/� D ŒtA� � Œ�A�. Let htbe the linear map from l.bIA/ into l.bI l.bIA// given by

ht .v ˝ a˝ w/ D cos.�t=2/2u˝ v ˝ a˝ w ˝ uC sin.�t=2/2v ˝ u˝ a˝ u˝ w� i cos.�t=2/ sin.�t=2/u˝ v ˝ a˝ u˝ wC i sin.�t=2/ cos.�t=2/v ˝ u˝ a˝ w ˝ u:

The family ht depends smoothly on t and we have h0 D i1 and h1 D i2. Since u isinvariant the map ht is in fact equivariant and one checks that ht is a homomorphism.Hence we have indeed defined a smooth homotopy between i1 and i2. This yieldsŒi1� D Œi2� and hence ŒtA� � Œ�A� D 1.

We derive the following general stability theorem.

Proposition 8.5 (Stability). Let H be a bornological quantum group and let A bean H -algebra. Moreover let V be an essential H -module and let b W V � V ! Cbe a nonzero equivariant bilinear pairing. Then there exists an invertible element inHPG0 .A; l.bIA//. Hence there are natural isomorphisms

HPH� .l.bIA/;B/ Š HPH� .A;B/ HPH� .A;B/ Š HPH� .A; l.bIB//for all H -algebras A and B .

Proof. Let us write ˇ W H � H ! C for the canonical equivariant bilinear pairingintroduced in Section 3. Moreover we denote by b� the pairing b W V� � V� ! Cwhere V� is the space V equipped with the trivial H -action. We have an equivariantisomorphism � W l.b� I l.ˇIA// Š l.ˇI l.bIA// given by

�.v ˝ .x ˝ a˝ y/˝ w/ D x.1/ ˝ x.2/ � v ˝ a˝ y.1/ � w ˝ y.2/

174 C. Voigt

and using ˇ.x; y/ D .S.y/x/ as well as the fact that is right invariant one checksthat � is an algebra homomorphism. Now we can apply Theorem 8.4 to obtain theassertion.

We deduce a simpler description of equivariant periodic cyclic homology in certaincases. A bornological quantum group H is said to be of compact type if the dualalgebra yH is unital. Moreover let us callH of semisimple type if it is of compact typeand the value of the integral for yH on 1 2 yH is nonzero. For instance, the dual of acosemisimple Hopf algebra yH is of semisimple type.

Proposition 8.6. Let H be a bornological quantum group of semisimple type. Thenwe have

HPH� .A;B/ Š H�.HomA.H/.XH .T A/;XH .T B///

for all H -algebras A and B .

Proof. Under the above assumptions the canonical bilinear pairing ˇ W yH � yH ! C isadmissible since the element 1 2 yH is invariant.

Finally we discuss excision in equivariant periodic cyclic homology. Consider anextension

K �� E� �� Q

ofH -algebras equipped with an equivariant linear splitting � W Q! E for the quotientmap � W E ! Q.

LetXH .T E W T Q/ be the kernel of the mapXH .T �/ W XH .T E/! XG.T Q/ in-duced by � . The splitting � yields a direct sum decompositionXH .T E/ D XH .T E WT Q/˚XH .T Q/ of AYD-modules. The resulting extension

XH .T E W T Q/ �� XH .T E/ �� XH .T Q/

of paracomplexes induces long exact sequences in homology in both variables. Thereis a natural covariant map � W XH .T K/! XH .T E W T Q/ of paracomplexes and wehave the following generalized excision theorem.

Theorem 8.7. The map � W XH .T K/! XH .T E W T Q/ is a homotopy equivalence.

This result implies excision in equivariant periodic cyclic homology.

Theorem 8.8 (Excision). Let A be an H -algebra and let .�; �/ W 0 ! K ! E !Q! 0 be an extension ofH -algebras with a bounded linear splitting. Then there aretwo natural exact sequences

HPH0 .A;K/��

��HPH0 .A;E/

�� HPH0 .A;Q/

��

HPH1 .A;Q/�� HPH1 .A;E/

�� HPH1 .A;K/


andHPH0 .Q;A/

��

��HPH0 .E;A/

�� HPH0 .K;A/

��

HPH1 .K;A/�� HPH1 .E;A/

�� HPH1 .Q;A/.

The horizontal maps in these diagrams are induced by the maps in the extension.

In Theorem 8.8 we only require a linear splitting for the quotient homomorphism� W E ! Q. Taking double crossed products of the extension given in Theorem 8.8yields an extension

K y KH�� E y KH

�� Q y KH

of H -algebras with a linear splitting. Using Lemma 3.3 one obtains an equivariantlinear splitting for this extension. Now we can apply Theorem 8.7 to this extension andobtain the claim by considering long exact sequences in homology.

For the proof of Theorem 8.7 one considers the left ideal L � T E generated byK �T E. The natural projection �E W T E ! E induces an equivariant homomorphism� W L! K and one obtains an extension

N �� L� �� K

of pro-H -algebras. The pro-H -algebraN is locally nilpotent and Theorem 8.7 followsfrom the following assertion.

Theorem 8.9. With the notations as above we have

a) The pro-H -algebra L is equivariantly quasifree.

b) The inclusion map L ! T E induces a homotopy equivalence W XH .L/ !XH .T E W T Q/.In order to prove the first part of Theorem 8.9 one constructs explicitly a projective

resolution of L of length one.

9 Comparison to previous approaches

In this section we discuss the relation of the theory developed above to the previousapproaches due to Akbarpour and Khalkhali [1], [2] as well as Neshveyev and Tuset[22]. As a natural domain in which all these theories are defined we choose to workwith actions of cosemisimple Hopf algebras over the complex numbers. Note that acosemisimple Hopf algebraH is a bornological quantum group with the fine bornology.In the sequel all vector spaces are equipped with the fine bornology when viewed asbornological vector spaces.

In their study of cyclic homology of crossed products [1] Akbarpour and Khalkhalidefine a cyclic module CH� .A/ of equivariant chains associated to a unital H -module

176 C. Voigt

algebraA. The space CHn .A/ in degree n of this cyclic module is the coinvariant spaceofH ˝A˝nC1 with respect to a certain action ofH . Recall that the coinvariant spaceVH associated to anH -module V is the quotient of V by the linear subspace generatedby all elements of the form t � v � �.t/v. Using the natural identification

nH .A/ D H ˝ A˝nC1 ˚H ˝ A˝n

the action considered by Akbarpour and Khalkhali corresponds precisely to the actionof H on the first summand in this decomposition. Hence CHn .A/ can be identifiedas a direct summand in the coinvariant space nH .A/H . Moreover, the cyclic modulestructure ofCH� .A/ reproduces the boundary operators b andB of H .A/H . We pointout that the relation T D id holds on the coinvariant space H .A/H which means thatthe latter is always a mixed complex.

It follows that there is a natural isomorphism of the cyclic type homologies associ-ated to the cyclic module CH� .A/ and the mixed complex H .A/H , respectively. Notealso that the complementary summand of CH� .A/ in H .A/H is obtained from thebar complex of A tensored with H . Since A is assumed to be a unital H -algebra, thiscomplementary summand is contractible with respect to the differential induced fromthe Hochschild boundary of H .A/H .

In the cohomological setting Akbarpour and Khalkhali introduce a cocyclic moduleC �H .A/ for every unital H -module algebra. The definition of this cocyclic modulegiven in [2] is not literally dual to the one of the cyclic module CH� .A/. In order toestablish the connection to our constructions let first A be an arbitrary H -algebra. Wedefine a modified action of H on H .A/ by the formula

t ı .x ˝ !/ D t.2/xS�1.t.3//˝ t.1/ � !and write H .A/� for the space H .A/ equipped with this action. Let us comparethe modified action with the original action

t � .x ˝ !/ D t.3/xS.t.1//˝ t.2/ � !introduced in Section 6. In the space H .A/H of coinvariants with respect to theoriginal action we have

t ı .x ˝ !/ D t.2/xS�1.t.3//˝ t.1/ � ! D t.4/xS�1.t.5//t.1/S.t.2//˝ t.3/ � !D t.2/ � .xS�1.t.3//t.1/ ˝ !/ D xS�1.t.2//t.1/ ˝ ! D �.t/ x ˝ !

which implies that the canonical projection H .A/ ! H .A/H factorizes over thecoinvariant space H .A/

�H with respect to the modified action. Similarly, in the coin-

variant space H .A/�H we have

t � .x ˝ !/ D t.3/xS.t.1//˝ t.2/ � ! D t.3/xS.t.1//t.5/S�1.t.4//˝ t.2/ � !D t.2/ ı .xS.t.1//t.3/ ˝ !/ D xS.t.1//t.2/ ˝ ! D �.t/ x ˝ !:

As a consequence we see that the identity map on H .A/ induces an isomorphism

H .A/H Š H .A/�H


between the coinvariant spaces.We may view a linear map H .A/ ! C as a linear map .A/ ! F.H/ where

F.H/ denotes the linear dual space of H . Under this identification an element inHomH . H .A/

�;C/ corresponds to a linear map f W .A/ ! F.H/ satisfying theequivariance condition

f .t � !/.x/ D f .!/.S.t.2//xt.1//for all t 2 H and! 2 .A/. In a completely analogous fashion to the case of homologydiscussed above, a direct inspection using the canonical isomorphisms

HomH . H .A/�;C/ Š Hom. H .A/

�H ;C/ Š Hom. H .A/H ;C/

shows that the cyclic type cohomologies of the cocyclic module C �H .A/ agree forevery unitalH -algebra A with the ones associated to the mixed complex H .A/H . Inparticular, the definition in [2] is indeed obtained by dualizing the construction givenin [1].

The main difference between the cocyclic module used byAkbarbour and Khalkhaliand the definition in [22] is that Neshveyev and Tuset work with right actions instead ofleft actions. It is explained in [22] that the two approaches lead to isomorphic cocyclicmodules and hence to isomorphic cyclic type cohomologies.

Now assume that H is a semisimple Hopf algebra. Then the coinvariant space H .A/H is naturally isomorphic to the space H .A/H of invariants. IfA is a unitalH -algebra then Theorem 6.9 and Proposition 8.6, together with the above considerations,yield a natural isomorphism

HP�.CH� .A// Š HPH� .C; A/which identifies the periodic cyclic homology of the cyclic module CH� .A/ with theequivariant cyclic homology of A in the sense of Definition 7.1. Similarly, for asemisimple Hopf algebra H one obtains a natural isomorphism

HP �.C �H .A// Š HPH� .A;C/for every unital H -algebra A.

Both of these isomorphisms fail to hold more generally, even in the classical settingof group actions. Let � be a discrete group and consider the group ringH D C� . Forthe H -algebra C with the trivial action one easily obtains

HP �.C �H .C// Š HomH .Had;C/

located in degree zero where Had is the space H D C� viewed as an H -module withthe adjoint action. On the other hand, a result in [25] shows that there is a naturalisomorphism

HPH� .C;C/ Š HP �.H/which identifies the H -equivariant theory of the complex numbers with the periodiccyclic cohomology of H D C� . This is the result one should expect from equivariant

178 C. Voigt

KK-theory [16]. Hence, roughly speaking, the theory in [2], [22] only recovers thedegree zero part of the group cohomology. A similar remark applies to the homologygroups defined in [1].

We mention that Akbarbour and Khalkhali have obtained an analogue of the Green–Julg isomorphism

HPH� .C; A/ Š HP�.A ÌH/if H is semisimple and A is a unital H -algebra [1]. This result holds in fact moregenerally in the case that H is the convolution algebra of a compact quantum groupand A is an arbitrary H -algebra. Similarly, there is a dual version

HPH� .A;C/ Š HP �.A ÌH/

of the Green–Julg theorem for the convolution algebras of discrete quantum groups.The latter generalizes the identification of equivariant periodic cyclic cohomology fordiscrete groups mentioned above. These results will be discussed elsewhere.

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A Schwartz type algebra for the tangent groupoid

Paulo Carrillo Rouse

1 Introduction

The concept of groupoid is central in non commutative geometry. Groupoids generalizethe concepts of spaces, groups and equivalence relations. It is clear nowadays thatgroupoids are natural substitutes of singular spaces. Many people have contributed torealizing this idea. We can find for instance a groupoid-like treatment in Dixmier’s workon transformation group, [11], or in Brown–Green–Rieffel’s work on orbit classificationof relations, [3]. In foliation theory, several models for the leaf space of a foliationwere realized using groupoids by several authors, for example by Haefliger ([12]) andWilkelnkemper ([24]), to mention some of them. There is also the case of orbifolds,these can be seen indeed as étale groupoids (see for example Moerdijk’s paper [17]).There are also some particular groupoid models for manifolds with corners and conicmanifolds, worked on by e.g. Monthubert [18], Debord–Lescure–Nistor ([10]) andAastrup–Melo–Monthubert–Schrohe ([1]).

The way we treat ”singular spaces” in non commutative geometry is by associatingto them algebras. In the case when the ”singular space” is represented by a Lie groupoid,we can, for instance, consider the convolution algebra of differentiable functions withcompact support over the groupoid (see Connes’ or Paterson’s books [8] and [22]).This last algebra plays the role of the algebra of smooth functions over the ”singularspace” represented by the groupoid. From the convolution algebra it is also possible toconstruct a C �-algebra, C �.G /, that plays, in some sense, the role of the algebra ofcontinuous functions over the ”singular space”. The idea of associating algebras in thissense can be traced back in works of Dixmier ([11]) for transformation groups, Connes([6]) for foliations and Renault ([23]) for locally compact groupoids, to mention someof them.

Using methods of non commutative geometry, we would like to get invariants ofthese algebras, and hence, of the spaces they represent. For that, Connes showedthat many groupoids and algebras associated to them appeared as ‘non commuta-tive analogues‘ of spaces to which many tools of geometry (and topology) such asK-theory and characteristic classes could be applied ([7], [8]). One classical way toobtain invariants in classical geometry (topology) is through the index theory in thesense of Atiyah–Singer. In the Lie groupoid case there is a pseudodifferential calculus,developed by Connes ([6]), Monthubert–Pierrot ([19]) and Nistor–Weinstein–Xu ([21])in general. Some interesting particular cases were treated in the groupoid-spirit by Mel-rose ([16]), Moroianu ([20]) and others (see [1]). Let G � G .0/ be a Lie groupoid.Then there is an analytic index morphism (see [19]),

inda W K0.A�G /! K0.C�.G //;

182 P. Carrillo Rouse

where AG is the Lie algebroid of G . The “C �-index” inda is a homotopy invariant ofthe G -pseudodifferential elliptic operators (G -PDO) and has proved to be very useful inmany different situations (see for example [2], [9], [10]). One way to define the aboveindex map is using Connes’ tangent groupoid associated to G as explained by Hilsumand Skandalis in [13] or by Monthubert and Pierrot in [19]. The tangent groupoid is aLie groupoid

G T � G .0/ � Œ0; 1�with G T WD AG �f0g t G � .0; 1�, and the groupoid structure is given by the groupoidstructure ofAG at t D 0 and by the groupoid structure of G for t ¤ 0. One of the mainfeatures of the tangent groupoid is that its C �-algebra C �.G T / is a continuous field ofC �-algebras over the closed interval Œ0; 1�, with associated fiber algebras

C0.A�G / at t D 0; and C �.G / for t ¤ 0.

In fact, it gives a C �-algebraic quantization of the Poisson manifold A�G (in the senseof [14]), and this is the main point why it allows to define the index morphism as asort of ”deformation”. Thus, the tangent groupoid construction has been very useful inindex theory ([2], [10], [13]), but also for other purposes ([14], [21]).

Now, to understand the purpose of the present work, let us first say that the indices (inthe sense of Atiyah–Singer–Connes) have not necessarily to be considered as elementsin K0.C �.G //. Indeed, it is possible to consider indices in K0.C1c .G //. The C1c -in-dices are more refined, but they have several inconveniences (see Alain Connes’ bookSection 9:ˇ for a discussion on this matter). Nevertheless this kind of indices have thegreat advantage that one can apply to them the existent tools (such as pairings withcyclic cocycles or Chern–Connes character) in order to obtain numerical invariants.

In this work we begin a study of more refined indices. In particular we are lookingfor indices between the C1c and the C �-levels, trying to keep the advantages of bothapproaches (see [5] for a more complete discussion). In the case of Lie groupoids,this refinement could mean to forget for a moment the powerful tools of the theory ofC �-algebras, and instead working in a purely algebraic and geometric level. In thepresent article, we construct an algebra of C1 functions over G T , denoted by Sc.G

T /.This algebra is also a field of algebras over the closed interval Œ0; 1�, with associatedfiber algebras

S.AG / at t D 0, and C1c .G / for t ¤ 0,

where S.AG / is the Schwartz algebra of the Lie algebroid. Furthermore, we will have

C1c .G T / � Sc.GT / � C �.G T /; (1)

as inclusions of algebras. Let us explain in some words why we define an algebra overthe tangent groupoid such that in zero it is Schwartz: The Schwartz algebras have ingeneral the good K-theory groups. For example we are interested in the symbols ofG -PDO, and, more precisely, in their homotopy classes in K-theory, that is, we areinterested in the group K0.A�G / D K0.C0.A

�G //. Here it would not be enough totake the K-theory of C1c .AG / (see the example [8], p. 142), however it is enough to

A Schwartz type algebra for the tangent groupoid 183

consider the Schwartz algebra S.A�G /. Indeed, the Fourier transform shows that thislast algebra is stable under holomorphic calculus on C0.A�G / and so it has the “good”K-theory, meaning that K0.A�G / D K0.S.A

�G //. None of the inclusions in (1) isstable under holomorphic calculus, but that is precisely what we wanted because ouralgebra Sc.G

T / has the remarkable property that its evaluation at zero is stable underholomorphic calculus, while its evaluation at one (for example) is not.

The algebra Sc.GT / is, as a vector space, a particular case of a more general

construction that we do for “deformation to the normal cone manifolds” from which thetangent groupoid is a special case (see [4] and [13]). A deformation to the normal conemanifold (DNC for short) is a manifold associated to an injective immersion X ,!M

that is considered as a sort of blow up in differential geometry. The constructionof a DNC manifold has very nice functorial properties (Section 3) which we exploitto achieve our construction. We think that our construction could be used also forother purposes, for example, it seems that it could help to give more understanding inquantization theory (see again [4]).

The article is organized as follows. In the second section we recall the basic factsabout Lie groupoids. We explain very briefly how to define the convolution algebraC1c .G /. In the third section we explain the “deformation to the normal cone” construc-tion associated to an injective immersion. Even if this could be considered as classicalmaterial, we do it in some detail since we will use in the sequel very explicit descriptionsthat we could not find elsewhere. We also review some functorial properties associatedto these deformations. A particular case of this construction is the tangent groupoid as-sociated to a Lie groupoid. In the fourth section we start by constructing a vector spaceSc.D

MX / for any deformation to the normal cone manifold DM

X ; this space alreadyexhibits the characteristic of being a field of vector spaces over the closed interval Œ0; 1�whose fiber at zero is a Schwartz space while for values different from zero it is equalto C1c .M/. We then define the algebra Sc.G

T /; the main result is that its product iswell-defined and associative. The last section is devoted to motivate the constructionof our algebra by explaining shortly some further developments that will immediatelyfollow from this work. All the results of the present work are part of the author’s PhDthesis.

I want to thank my PhD advisor, Georges Skandalis, for all the ideas that heshared with me, and for all the comments and remarks he made on the present work.I would also like to thank the referee for the useful comments he made for improvingthis paper.

2 Lie groupoids

Let us recall what a groupoid is.

Definition 2.1. A groupoid consists of the following data: two sets G and G .0/, andmaps

• s; r W G ! G .0/, called the source and target map respectively,


• m W G .2/ ! G , called the product map (where G .2/ D f.�; �/ 2 G � G W s.�/ Dr.�/g),

• u W G .0/ ! G the unit map, and

• i W G ! G the inverse map

such that, if we write m.�; �/ D � � �, u.x/ D x and i.�/ D ��1, we have

1. � � .� � ı/ D .� � �/ � ı for all �; �; ı 2 G when this is possible,

2. � � x D � and x � � D � for all �; � 2 G with s.�/ D x and r.�/ D x,

3. � � ��1 D u.r.�// and ��1 � � D u.s.�// for all � 2 G ,

4. r.� � �/ D r.�/ and s.� � �/ D s.�/.Generally, we denote a groupoid by G � G .0/ where the parallel arrows are the sourceand target maps and the other maps are given.

Now, a Lie groupoid is a groupoid in which every set and map appearing inthe last definition is C1 (possibly with borders), and the source and target mapsare submersions. For A;B subsets of G .0/ we use the notation GBA for the subsetf� 2 G W s.�/ 2 A; r.�/ 2 Bg.

Throughout this paper, G � G .0/ is a Lie groupoid. We recall how to define analgebra structure in C1c .G / using smooth Haar systems.

Definition 2.2. A smooth Haar system over a Lie groupoid consists of a family ofmeasures �x in Gx for each x 2 G .0/ such that

• for � 2 Gyx we have the following compatibility condition:

ZGx

f .�/ d�x.�/ DZ

Gy

f .� ı �/ d�y.�/;

• for each f 2 C1c .G / the map

x 7!Z

Gx

f .�/ d�x.�/

belongs to C1c .G .0//.

A Lie groupoid always possesses a smooth Haar system. In fact, if we fix a smooth(positive) section of the 1-density bundle associated to the Lie algebroid we obtain asmooth Haar system in a canonical way. The advantage of using 1-densities is that themeasures are locally equivalent to the Lebesgue measure. We suppose for the rest ofthe paper an underlying smooth Haar system given by 1-densities (for complete details


see [22]). We can now define a convolution product on C1c .G /. For f; g 2 C1c .G /we set

.f � g/.�/ DZ

Gs.�/

f .� � ��1/g.�/ d�s.�/.�/:

This gives a well-defined associative product.

Remark 2.3. There is a way to avoid the Haar system when one works with Liegroupoids, using half densities (see Connes’ book [8]).

3 Deformation to the normal cone

Let M be a C1 manifold and X �M be a C1 submanifold. We denote by NMX the

normal bundle to X in M , i:e:, NMX WD TXM=TX . We define the following set

DMX WD NM

X � 0 tM � .0; 1�: (2)

The purpose of this section is to recall how to define a C1-structure with boundary inDMX . This is more or less classical, for example it was extensively used in [13]. Here

we are only going to do a sketch.Let us first consider the case whereM D Rn andX D Rp�f0g (where we identify

canonically X D Rp). We denote by q D n�p and by Dnp for DRn

Rp as above. In thiscase we clearly have that Dn

p D Rp �Rq � Œ0; 1� (as a set). Consider the bijection

‰ W Rp �Rq � Œ0; 1�! Dnp (3)

given by

‰.x; �; t/ D´.x; �; 0/ if t D 0;.x; t�; t/ if t > 0;

whose the inverse is given explicitly by

‰�1.x; �; t/ D´.x; �; 0/ if t D 0;.x; 1

t�; t/ if t > 0:

We can consider the C1-structure with border on Dnp induced by this bijection.

In the general case, let .U; �/ be a local chart in M and suppose it is an X -slice,so that it satisfies

1) � W U Š! U � Rp �Rq;

2) if V D U \X , V D U \ .Rp � 0/, then we have V D ��1.V /.With this notation we have that DU

V � Dnp is an open subset. We may define a function

Q� W DUV ! DU

V


in the following way. For x 2 V we have �.x/ 2 Rp � f0g. If we write �.x/ D.�1.x/; 0/, then

�1 W V ! V � Rp

is a diffeomorphism, whereV D U \.Rp�f0g/. Set Q�.v; �; 0/ D .�1.v/; dN�v.�/; 0/and Q�.u; t/ D .�.u/; t/ for t ¤ 0. Here dN�v W Nv ! Rq is the normal componentof the derivative d�v for v 2 V . It is clear that Q� is also a bijection (in particular itinduces a C1 structure with border over DU

V).

Let us define, with the same notations as above, the following set:

�UV D f.x; �; t/ 2 Rp �Rq � Œ0; 1� W .x; t � �/ 2 U g:This is an open subset of Rp �Rq � Œ0; 1� and thus a C1 manifold (with border). It isimmediate that DU

V is diffeomorphic to �UV through the restriction of ‰ given in (3).Now we consider an atlas f.U˛; �˛/g˛2� of M consisting of X -slices. It is clear that

DMX D

[˛2�

DU˛

V˛(4)

and if we take DU˛

V˛

'˛! �U˛V˛

defined as the composition

DU˛

V˛

�˛! DU˛V˛

‰�1˛! �

U˛V˛

then we obtain the following result.

Proposition 3.1. f.DU˛

V˛; '˛/g˛2� is a C1 atlas with border over DM

X .

In fact the proposition can be proved directly from the following elementary lemma.

Lemma 3.2. Let F W U ! U 0 be a C1 diffeomorphism where U � Rp � Rq andU 0 � Rp � Rq are open subsets. We write F D .F1; F2/ and we suppose thatF2.x; 0/ D 0. Then the function QF W �UV ! �U

0

V 0 defined by

QF .x; �; t/ D´.F1.x; 0/;

@F2@�.x; 0/ � �; 0/ if t D 0;

.F1.x; t�/;1tF2.x; t�/; t/ if t > 0:

is a C1 map.

Proof. Since the result will hold if and only if it is true in each coordinate, it is enoughto prove that if we have a C1 map F W U ! R with F.x; 0/ D 0, then the mapQF W �UV ! R given by

QF .x; �; t/ D´@F@�.x; 0/ � � if t D 0;

1tF.x; t�/ if t > 0

is a C1 map. For that we write

F.x; �/ D @F

@�.x; 0/ � � C h.x; �/ � �


with h W U ! Rq a C1 map such that h.x; 0/ D 0. Then

1

tF.x; t�/ D @F

@�.x; 0/ � � C h.x; t�/ � �

from which we immediately get the result.

Definition 3.3 (DNC). Let X � M be as above. The set DMX provided with the C1

structure with border induced by the atlas described in the last proposition is called thedeformation to normal cone associated to X � M . We will often write DNC insteadof deformation to the normal cone.

Remark 3.4. Following the same steps, it is possible to define a deformation to thenormal cone associated to an injective immersion X ,!M .

Let us mention some basic examples of DCN manifolds DMX .

Examples 3.5. 1. Consider the case when X D ;. We have that DM; D M � .0; 1�with the usual C1 structure on M � .0; 1�. We used this fact implicitly to cover DM

X

as in (4).2. Consider the case when X � M is an open subset. Then we do not have any

deformation at zero and we immediately see from the definition that DMX is just the

open subset of M � Œ0; 1� consisting of the union of X � Œ0; 1� and M � .0; 1�.The most important feature of the DNC construction is that it is in some sense

functorial. More explicitly, let .M;X/ and .M 0; X 0/ be C1-couples as above and letF W .M;X/ ! .M 0; X 0/ be a couple morphism, i.e., a C1 map F W M ! M 0 withF.X/ � X 0. We define D.F / W DM

X ! DM 0

X 0 by the following formulas:

D.F /.x; �; 0/ D .F.x/; dNFx.�/; 0/ and D.F /.m; t/ D .F.m/; t/ for t ¤ 0;where dNFx is by definition the map

.NMX /x

dNFx�! .NM 0

X 0 /F.x/

induced by TxMdFx�! TF.x/M

0.We have the following proposition, which is also an immediate consequence of the

lemma above.

Proposition 3.6. The map D.F / W DMX ! DM 0

X 0 is C1.

Remark 3.7. If we consider the category C12 ofC1 pairs given by aC1manifold anda C1 submanifold and pair morphisms as above, we can reformulate the propositionand say that we have a functor

D W C12 ! C1

where C1 denotes the category of C1 manifolds with border.


3.1 The tangent groupoid

Definition 3.8 (Tangent groupoid). Let G � G .0/ be a Lie groupoid. The tangentgroupoid associated to G is the groupoid that has DG

G .0/as the set of arrows and

G .0/ � Œ0; 1� as the units with

• sT .x; �; 0/ D .x; 0/ and rT .x; �; 0/ D .x; 0/ at t D 0,

• sT .�; t/ D .s.�/; t/ and rT .�; t/ D .r.�/; t/ at t ¤ 0,

• the product is given by mT ..x; �; 0/; .x; �; 0// D .x; � C �; 0/ andmT ..�; t/; .ˇ; t// D .m.�; ˇ/; t/ if t ¤ 0 and if r.ˇ/ D s.�/,

• the unit map uT W G .0/ ! G T is given by uT .x; 0/ D .x; 0/ and uT .x; t/ D.u.x/; t/ for t ¤ 0.

We denote G T WD DGG .0/

.

As we have seen above G T can be considered as a C1 manifold with border. As aconsequence of the functoriality of the DNC construction we can show that the tangentgroupoid is in fact a Lie groupoid. Indeed, it is easy to check that if we identify in acanonical way DG .2/

G .0/with .G T /.2/, then

mT D D.m/; sT D D.s/; rT D D.r/; uT D D.u/

where we are considering the following pair morphisms:

m W ..G /.2/;G .0//! .G ;G .0//;

s; r W .G ;G .0//! .G .0/;G .0//;

u W .G .0/;G .0//! .G ;G .0//:

Finally, if f�xg is a smooth Haar system on G , then, setting

• �.x;0/ WD �x at .G T /.x;0/ D TxGx , and

• �.x;t/ WD t�q � �x at .G T /.x;t/ D Gx for t ¤ 0, where q D dim Gx ,

one obtains a smooth Haar system for the tangent groupoid (details may be found in[22]).

We finish this section with some interesting examples of groupoids and their tangentgroupoids.

Examples 3.9. 1. The tangent groupoid of a group. Let G be a Lie group consideredas a Lie groupoid, G WD G � feg. In this case the normal bundle to the inclusionfeg ,! G is of course identified with the Lie algebra of the group. Hence, the tangentgroupoid is a deformation of the group in its Lie algebra:

G T D g � f0g t G � .0; 1�:


2. The tangent groupoid of a smooth vector bundle. Let Ep! X be a smooth

vector bundle over a (connected) C1 manifold X . We can consider the Lie groupoidE � X induced by the vector structure of the fibers, i:e:, s.�/ D p.�/ D r.�/ and thecomposition is given by the vector sum � ı � D � C �. In this case the normal vectorbundle associated to the zero section can be identified to E itself. Hence, as a set thetangent groupoid is E � Œ0; 1�, but the C1-structure at zero is given locally as in (3).

3. The tangent groupoid of a C1-manifold. Let M a C1-manifold. We canconsider the product groupoid GM WD M �M � M . The tangent groupoid in thiscase takes the following form:

G TM D TM � f0g t M �M � .0; 1�:This is called the tangent groupoid to M and it was introduced by Connes for giving avery conceptual proof of the Atiyah–Singer index theorem (see [8] and [10]).

4 An algebra for the tangent groupoid

In this section we will show how to construct an algebra for the tangent groupoid whichconsists of C1 functions that satisfy a rapid decay condition at zero while out of zerothey satisfy a compact support condition. This algebra is the main construction in thiswork.

4.1 Schwartz type spaces for deformation to the normal cone manifolds. Ouralgebra for the tangent groupoid will be a particular case of a construction associated toany deformation to the normal cone. We start by defining a space for DNCs associatedto open subsets of Rp �Rq .

Definition 4.1. Let p; q 2 N and U � Rp � Rq an open subset, and let V DU \ .Rp � f0g/.

(1) LetK � U � Œ0; 1� be a compact subset. We say thatK is a conic compact subsetof U � Œ0; 1� relative to V if

K0 D K \ .U � f0g/ � V:(2) Let g 2 C1.�UV /. We say that f has compact conic support K, if there exists a

conic compactK of U � Œ0; 1� relative to V such that if t ¤ 0 and .x; t�; t/ … Kthen g.x; �; t/ D 0.

(3) We denote by Sc.�UV / the set of functions g 2 C1.�UV / that have compact conic

support and that satisfy the following condition:

.s1) for all k;m 2 N, l 2 Np and ˛ 2 Nq there exists C.k;m;l;˛/ > 0 such that

.1C k�k2/kk@lx@˛� @mt g.x; �; t/k � C.k;m;l;˛/:


Now, the spaces Sc.�UV / are invariant under diffeomorphisms. More precisely if

F W U ! U 0 is a C1 diffeomorphism as in Lemma 3.2 then we can prove the nextresult.

Proposition 4.2. Let g 2 Sc.�U 0

V 0 /, then Qg WD g ı QF 2 Sc.�UV /.

Proof. The first observation is that Qg 2 C1.�UV /, thanks to Lemma 3.2. Let us checkthat it has compact conic support. For that, let K 0 � U 0 � Œ0; 1� be the conic compactsupport of g. We let

K D .F �1 � IdŒ0;1�/ � U � Œ0; 1�;which is a conic compact subset of U � Œ0; 1� relative to V , and it is immediate bydefinition that Qg.x; �; t/ D 0 if t ¤ 0 and .x; t � �; t/ … K, that is, Qg has compact conicsupport K.

We now check the rapid decay property .s1/: To simplify the proof we first in-troduce some useful notation. Writing F D .F1; F2/ as in Lemma 3.2, we de-note F1.x; �/ D .A1.x; �/; : : : ; Ap.x; �// and F2.x; �/ D .B1.x; �/; : : : ; Bq.x; �//.We also write w D w.x; �; t/ D .A1.x; t�/; : : : ; Ap.x; t�// and � D �.x; �; t/ D. QB1.x; �; t/; : : : ; QBq.x; �; t// where QBj is as above, i:e:,

QBj .x; �; t/ D

8<:@Bj@�.x; 0/ � � if t D 0;

1tBj .x; t�/ if t ¤ 0:

In particular by definition we have QF .x; �; t/ D .w; �; t/. We also write z D .x; �; t/

and u D .!; �; t/. Hence, what we would like is to find bounds for expressions of thefollowing type:

k�kkk@˛z Qg.z/kfor arbitrary k 2 N and ˛ 2 Np � Nq � N. A simple calculation shows that thederivatives @˛z Qg.z/ are of the following form:

@˛z Qg.z/ DXjˇ j�j˛j

Pˇ .z/@ˇug.u/

where Pˇ .z/ is a finite sum of products of the form

@�z!i .z/ � @ız�j .z/:We are only interested to see what happens in the set K� WD fz D .x; �; t/ 2 � W.x; t � �; t/ 2 Kg since outside of this set we have that g and all its derivatives vanish(.x; t�; t/ 2 K iff .w; t�; t/ 2 K 0). For a point z D .x; �; t/ 2 K� we have that.x; t � �/ is contained in a compact set, and then it follows that the expressions

k@�z!i .z/k


are bounded in K�. For the expressions k@ız�j .z/k, we proceed first by developing asin Lemma 3.2, that is,

�j .x; �; t/ D�@Bj

@�.x; 0/ � � C hj .x; t�/

�� :

Now, since we are only considering points in K�, it is immediate that we can findconstants Cj > 0 such that

k@ız�j .z/k � Cj � k�kmı :In the same way (remember that F is a diffeomorphism) we can have constants Ci > 0such that

k�i .!; �; t/k � Ci � k�k:Putting all together, and using the property .s1/ for g, we get bounds C > 0 such that

k�kkk@˛z Qg.z/k � C;and this concludes the proof.

Remark 4.3. We can resume the last invariance result as follows: If .U;V/ is aC1 pair diffeomorphic to .U; V / with U � E, an open subset of a vector spaceE, and V D U \ E, then Sc.D

UV/ is well defined and does not depend on the pair

diffeomorphism.

With the last compatibility result in hand we are ready to give the main definitionin this work.

Definition 4.4. Let g 2 C1.DMX /.

(a) We say that g has compact conic support K, if there exists a compact subsetK �M � Œ0; 1� withK0 WD K \ .M � f0g/ � X (conic compact relative to X )such that if t ¤ 0 and .m; t/ … K then g.m; t/ D 0.

(b) We say that g is rapidly decaying at zero if for every X -slice chart .U; �/ andfor every 2 C1c .U � Œ0; 1�/, the map g� 2 C1.�UV / given by

g�.x; �; t/ D .g ı '�1/.x; �; t/ � . ı p ı '�1/.x; �; t/is in Sc.�

UV /, where p is the projection p W DM

X ! M � Œ0; 1� given by.x; �; 0/ 7! .x; 0/, and .m; t/ 7! .m; t/ for t ¤ 0.

Finally, we denote by Sc.DMX / the set of functions g 2 C1.DM

X / that are rapidlydecaying at zero with compact conic support.

Remark 4.5. (a) It follows from the definition of Sc.DMX / that the latter contains

C1c .DMX / as a vector subspace.

(b) It is clear that C1c .M � .0; 1�/ can be considered as a subspace of Sc.DMX / by

extending by zero the functions at NMX .


Following the lines of the last remark we are going to precise a possible decompo-sition of our space Sc.D

MX / that will be very useful in the sequel. Let f.U˛; �˛/g˛2�

a family of X -slices covering X . Consider the open cover of M � Œ0; 1� consisting off.U˛ � Œ0; 1�; �˛/g˛2� together with M � .0; 1�. We can take a partition of the unitysubordinated to the last cover,

f˛; g˛2�:That is, we have the following properties:

• 0 � ˛; � 1,

• supp˛ � U˛ � Œ0; 1� and supp �M � .0; 1�,•

P˛ ˛ C

P D 1.

Let f 2 Sc.DMX /. If we denote

f˛ WD f jDU˛V˛

� .˛ ı p/ 2 C1.DU˛

V˛/

andf WD f jM�.0;1� � . ı p/ 2 C1.M � .0; 1�/;

then we obtain the following decomposition:

f DX˛

f˛ C f:

Now, since f is conic compactly supported we can suppose, without lost of generality,that

• f 2 C1c .M � .0; 1�/, and

• that ˛ is compactly supported in U˛ � Œ0; 1�.What we conclude of all this, is that we can decompose our space Sc.D

MX / as follows:

Sc.DMX / D

X˛2ƒ

Sc.DU˛

V˛/C C1c .M � .0; 1�/: (5)

As we mentioned in the introduction, we want to see the space Sc.DMX / as a field

of vector spaces over the interval Œ0; 1�, where at zero we talk about Schwartz spaces.In our case we are interested in Schwartz functions on the vector bundle NM

X . Let usfirst recall the notion of the Schwartz space associated to a vector bundle.

Definition 4.6. Let .E; p;X/ be a smooth vector bundle over a C1 manifold X . Wedefine the Schwartz space S.E/ as the set ofC1 functions g 2 C1.E/ such that g is aSchwartz function at each fiber (uniformly) and g has compact support in the directionof X , i:e:, if there exists a compact subset K � X such that g.Ex/ D 0 for x … K.


The vector space S.E/ is an associative algebra with the product given as follows:for f; g 2 S.E/, we put

.f � g/.�/ DZEp.�/

f .� � �/g.�/ d�p.�/.�/; (6)

where �� is a smooth Haar system of the Lie groupoid E � X . A classical Fourierargument can be applied to show that the algebra S.E/ is isomorphic to .S.E�/; �/(punctual product). In particular this implies that K0.S.E// Š K0.E�/.

In the case we are interested in, we have a couple .M;X/ and a vector bundleassociated to it, that is, the normal bundle over X , NM

X . The reason why we gave thelast definition is because we get evaluation linear maps

e0 W Sc.DMX /! S.NM

X /; (7)

and

et W Sc.DMX /! C1c .M/ (8)

for t ¤ 0. Consequently, we have that the vector space Sc.DMX / is a field of vector

spaces over the closed interval Œ0; 1� whose fiber spaces are S.NMX / at t D 0 and

C1c .M/ for t ¤ 0.Let us finish this subsection by giving the examples of spaces Sc.D

MX / correspond-

ing to the DCN manifolds in Examples 3.5.

Examples 4.7. 1. For X D ;, we have that Sc.DM; / Š C1c .M � .0; 1�/.

2. For X � M an open subset we have that Sc.DMX / Š C1c .W / where W �

M � Œ0; 1� is the open subset consisting of the union of X � Œ0; 1� and M � .0; 1�.

4.2 Schwartz type algebra for the tangent groupoid. In this section we define analgebra structure on Sc.G

T /. We start by defining a function mc W Sc.DG .2/

G .0// !

Sc.DGG .0/

/ by the following formulas:

For F 2 Sc.DG .2/

G .0//, we let

mc.F /.x; �; 0/ DZTxGx

F.x; � � �; �; 0/ d�x.�/

and

mc.F /.�; t/ DZ

Gs.�/

F.� ı ı�1; ı; t/t�q d�s.�/.ı/:

If we canonically identify DG .2/

G .0/with .G T /.2/, the map above is nothing else than the

integration along the fibers ofmT W .G T /.2/ ! G T . We have the following proposition:

Proposition 4.8. mc W Sc..G T /.2//! Sc.GT / is a well-defined linear map.


The interesting part of the proposition is that the map is well defined since it isevidently be linear. Let us suppose for the moment that the proposition is true. Underthis assumption, we will define the product in Sc.G

T /.

Definition 4.9. Let f; g 2 Sc.GT /, we define a function f � g in G T by

.f � g/.x; �; 0/ DZTxGx

f .x; � � �; 0/g.x; �; 0/ d�x.�/

and

.f � g/.�; t/ DZ

Gs.�/

f .� ı ı�1; t /g.ı; t/t�q d�s.�/.ı/

for t ¤ 0.

We can enounce our main result.

Theorem 4.10. � defines an associative product on Sc.GT /.

Proof. Remember that we are assuming for the moment the validity of Proposition 4.8.Let f; g 2 Sc.G

T /. We let F WD .f; g/ be the function in .G T /.2/ defined by

.f; g/.x; �; �; 0/ D f .x; �; 0/ � g.x; �; 0/and

.f; g/..�; t/; .ı; t// D f .�; t/ � g.ı; t/for t ¤ 0. Now, from the Leibnitz formula for the derivative of a product it is immediatethat .f; g/ 2 Sc..G

T /.2//. Finally, by definition we have that

mc..f; g// D f � g;hence, by Proposition 4.8, f � g is a well-defined element in Sc.G

T /.For the associativity of the product, let us remark that when one restricts the product

toC1c .G T /, this coincides with the product classically considered onC1c .G T / (whichis associative, see for example [22]). The associativity for Sc.G

T / is proved exactly inthe same way that for C1c .G T /.

It remains to prove Proposition 4.8. We are going to start locally. Let U 2 Rp �Rq�Rq be an open set andV D U \Rp�f0g�f0g. LetP W Rp�Rq�Rq ! Rp�Rq

the canonical projection .x; �; �/ 7! .x; �/. We set U 0 D P.U / 2 Rp � Rq , then U 0is also an open subset, V Š U 0 \Rp � f0g and P jV D IdV . We denote also by P therestriction P W U ! U 0. As in Lemma 3.2 we have a C1 map QP W �UV ! �U

0

V whichin this case is explicitly written as

QP .x; �; �; t/ D .x; �; t/:We define QPc W Sc.�UV /! Sc.�

U 0

V / as follows:

QPc.F /.x; �; t/ DZf�2Rq W.x;;�;t/2�U

VgF.x; �; �; t/ d�:

Let us prove a lemma.


Lemma 4.11. QPc W Sc.�UV /! Sc.�U 0

V / is well defined.

Proof. The first observation is that the integral in the definition of QPc is always welldefined. Indeed, this is a consequence of the following two facts:

• for t D 0, � 7! F.x; �; �; 0/ 2 S.Rq/,

• for t ¤ 0, � 7! F.x; �; �; t/ 2 C1c .Rq/.Taking derivatives under the integral sign, we obtain that QPc.F / 2 C1.�U 0

V /. Then,we just have to show that QPc.F / verifies the two conditions of Definition 4.1. For thefirst, if K � U � Œ0; 1� is the compact conic support of F , then it is enough to put

K 0 D .P � IdŒ0;1�/.K/

in order to obtain a conic compact subset of U 0 � Œ0; 1� relative to V and to check thatK 0 is the compact conic support of QPc.F / . Let us now verify the condition .s1/. Letk;m 2 N, l 2 Np and ˇ 2 Nq . We want to find C.k;m;l;ˇ/ > 0 such that

.1C k�k2/kk@lx@ˇ @mt QPc.F /.x; �; t/k � C.k;m;l;˛/:For k0 � k C q

2and ˛ D .0; ˇ/ 2 Rq � Rq we have by hypothesis that there exists

C 0.k0;m;l;˛/

> 0 such that

k@lx@ˇ @mt F.x; �; �; t/k � C 01

.1C k.�; �/k2/k0:

Then we also have that

k@lx@ˇ @mt QPc.F /.x; �; t/k � C 0Zf�2Rq W.x;;�;t/2�U

Vg

1

.1C k.�; �/k2/k0d�

� C 0 1

.1C k�k2/ q2�k0

Zf�2Rqg

1

.1C k�k2/k0d� � C 1

.1C k�k2/kwith

C D C 0 �Zf�2Rqg

1

.1C k�k2/k0d�:

We can now give the proof of Proposition 4.8.

Proof of Proposition 4.8. Let us first fix some notation. We suppose dim G D p C qand dim G .0/ D p, in particular this implies that dim G .2/ D p C q C q. Let .U; �/and .U0; �0/ be G .0/-slices in G .2/ and G , respectively, such that the following diagramcommutes:

Um ��

�

��

U0

�0

��

UP

�� U 0,


where P W Rp � Rq � Rq ! Rp � Rq is the canonical projection (as above) andP.U / D U 0. This is possible since m is a surjective submersion. Now, we apply theDNC construction to the diagram above to obtain, thanks to the functoriality of theconstruction, the following commutative diagram:

DUV

D.m/��

Q��

DU0

V 0

Q�0

��

DUV D.P /

�� DU 0

V 0

��

‰�

��

QP�� 0 ,

‰�0

��

where ‰� and Q� are as in Section 3. Let g 2 Sc.DUV / and define

Pc.g/.x; �; t/ D

8<:

RRq g.x; �; �; 0/ d� if t D 0;

Rf�2Rq W.x;;�;t/2�U

Vg g.x; �; �; t/t

�q d� if t ¤ 0:Then, from the last commutative diagram, we get that

Pc.g/ D QPc.g ı‰�/ ı .‰�0/�1;

hence, thanks to Lemma 4.11, we can conclude that we have a well-defined linear map

Pc W Sc.DUV /! Sc.D

U 0

V 0 /:

We now use the Proposition 4.2 to write

Sc.DUV / D fh 2 C1.DU

V / W h ı Q��1 2 Sc.DUV /g;

and so for h 2 Sc.DUV/ we see that

Pc.h ı Q��1/ ı Q�0 2 Sc.DU0

V 0 /:

We use again the second commutative diagram to see that

mc.h/ D Pc.h ı Q��1/ ı Q�0:We then have a well-defined linear map

mc W Sc.DUV /! Sc.D

U0

V 0 /:

To pass to the global case we only have to use the decomposition of Sc.DG .2/

G .0// and

of Sc.DGG .0/

/ as in (5), and of course the invariance under diffeomorphisms (Proposi-tion 4.2).


Recall that we have well-defined evaluation morphisms as in (7) and (8). In thecase of the tangent groupoid they are by definition morphisms of algebras. Hence, thealgebra Sc.G

T / is a field of algebras over the closed interval Œ0; 1�, with associatedfiber algebras,

S.AG / at t D 0, and C1c .G / for t ¤ 0.

It is very interesting to see what this means in the examples given in 3.9.

5 Further developments

Let G � G .0/ be a Lie groupoid. In index theory for Lie groupoids the tangent groupoidhas been used to define the analytic index associated to the group, as a morphismK0.A�G / ! K0.C

�r .G // (see [19]) or as a KK-element in KK.C0.A�G /; C �r .G //

(see [13]), this can be done because one has the following short exact sequence ofC �-algebras

0! C �r .G � .0; 1�/ �! C �r .G T /e0�! C0.A

�G / �! 0; (9)

and because of the fact that theK-groups of the algebra C �r .G � .0; 1�/ vanish (homo-topy invariance). The index defined at the C �-level has proven to be very useful (seefor example [9]) but extracting numerical invariants from it, with the existent tools, isvery difficult. In non commutative geometry, and also in classical geometry, the toolsfor obtaining more explicit invariants are more developed for the ’smooth objects’; inour case this means the convolution algebra C1c .G /, where we can for example applyChern–Weil–Connes theory. Hence, in some way, the indices defined in K0.C1c .G //are more refined objects and for some cases it would be preferable to work with them.Unfortunately these indices are not good enough, since for example they are not homo-topy invariant; in [8] Alain Connes discusses this and also other reasons why it is notenough to keep with theC1c -indices. The main reason to construct the algebra Sc.G

T /

is that it gives an intermediate way between the C1c -level and the C �-level and willallow us in [5] to define another analytic index morphism associated to the groupoid,with the advantage that this index will take values in a group that allows to do pairingswith cyclic cocycles and in general to apply Chern–Connes theory to it. The way weare going to define our index is by obtaining first a short exact sequence analogous to(9), that is, a sequence of the following kind

0! J �! Sc.GT /

e0�! S.A�G // �! 0: (10)

The problem here will be that we do not dispose of the advantages of the K-theory forC �-algebras, since the algebras we are considering are not of this type (we do not havefor example homotopy invariance).

References

[1] J. Aastrup, S. T. Melo, , B. Monthubert, E. Schrohe, Boutet de Monvel’s Calculus andGroupoids I, Preprint 2006, arxiv:math.KT/0611336.


[2] J. Aastrup, R. Nest, and E. Schrohe, A continuous field of C�-algebras and the tangentgroupoid for manifolds with boundary, J. Funct. Anal. 237 (2006), 482–506.

[3] L. Brown, P. Green, and M. Rieffel, Stable isomorphism and stable Morita equivalence forC�-algebras, Pacific J. Math. 71 (1977), 349–363.

[4] J. Carinena, J. Clemente-Gallardo, E. Follana, J. M. Gracia-Bondia, A. Rivero, and J. C.Varilly, Connes’ tangent groupoid and strict quantization, J. Geom. Phys. 32 (1999), 79–96.

[5] P. Carrillo, An analytic index for Lie groupoids, Preprint 2006, arxiv:math.KT/0612455.

[6] A. Connes, Sur la théorie non commutative de l’intégration, in Algèbres d’opérateurs,Lecture Notes in Math. 725, Springer-Verlag, Berlin 1979, 19–143.

[7] A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62(1985), 257–360.

[8] A. Connes, Non commutative geometry, Academic Press, San Diego, CA, 1994.

[9] A. Connes, and G. Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst.Math 6 (1984), 1139–1183.

[10] C. Debord, J. M. Lescure, and V. Nistor, Groupoids and an index theorem for conicalpseudomanifolds, Preprint 2008, arxiv:math.OA/0609438.

[11] J. Dixmier, Les C�-algèbres et leurs représentations, Cahiers Scientifiques XXIX,Gauthier-Villars, Paris 1964.

[12] A. Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque 116 (1982), 70–97.

[13] M. Hilsum and G. Skandalis, MorphismesK-orientés d’espaces de feuilles et fonctorialitéen théorie de Kasparov (d’après une conjecture d’A. Connes), Ann. Sci. École Norm. Sup.(4) 20 (1987), 325–390.

[14] N. P. Landsman, Quantization and the tangent groupoid, in Operator algebras and mathe-matical physics, Theta, Bucharest 2003, 251–265.

[15] R. Lauter, B. Monthubert, and V. Nistor, Pseudodifferential analysis on continuous familygroupoids, Doc. Math 5 (2000), 625–655.

[16] R. Melrose, The Atiyah-Patodi-Singer index theorem, Res. Notes in Math. 4, AK Peters,Wellesley, MA, 1993.

[17] I. Moerdijk, Orbifolds as groupoids: an introduction, in Orbifolds in mathematics andphysics (Madison, WI, 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, RI,205–222.

[18] B. Monthubert, Groupoids and Pseudodifferential calculus on manifolds with corners, J.Funct. Anal. 199 (2003), 243–286.

[19] B. Monthubert and F. Pierrot, Indice analytique et groupoïdes de Lie, C. R. Acad. Sci. Paris325 (1997), 193–198.

[20] S. Moroinau, Adiabatic limits of eta and zeta functions of elliptic operators, Math. Z. 246(2004), 441–471.

[21] V. Nistor, A. Weinstein, P. and Xu, Pseudodifferential operators on differential groupoids,Pacific J. Math 189 (1999), 117–152.

[22] A. Paterson, Groupoids and inverse semigroups, and their operator algebras, Progr. Math.170, Birkhäuser, Boston, MA, 1999.


[23] J. Renault, A groupoid approach to C�-algebras, Lecture Notes in Math. 793, Springer-Verlag, Berlin 1980.

[24] H. E. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1 (1983), 51–75.

C �-algebras associated with the ax C b-semigroupover N

Joachim Cuntz�

1 Introduction

In this note we present aC �-algebra (denoted by QN) which is associated to the axCb-semigroup over N. It is in fact a natural quotient of the C �-algebra associated withthis semigroup by some additional relations (which make it simple and purely infinite).These relations are satisfied in representations related to number theory. The studyof this algebra is motivated by the construction of Bost–Connes in [1]. Our C �-al-gebra contains the algebra considered by Bost–Connes, but in addition a generatorcorresponding to translation by the additive group Z.

As a C �-algebra, QN has an interesting structure. It is a crossed product of theBunce–Deddens algebra associated to Q by the action of the multiplicative semigroupN�. It has a unique canonical KMS-state. We also determine its K-theory, whosegenerators turn out to be determined by prime numbers.

On the other hand, QN can also be obtained as a crossed product of the commutativealgebra of continuous functions on the completion OZ by the natural action of the axCb-semigroup over N. More interestingly, its stabilization is isomorphic to the crossedproduct of the algebra C0.Af / of continuous functions on the space of finite adeles bythe natural action of the axCb-group PCQ over Q. Somewhat surprisingly one obtainsexactly the same C �-algebra QN (up to stabilization) working with the completion Rof Q at the infinite place and taking the crossed product by the natural action of theax C b-group PCQ on C0.R/.

In the last section we consider the analogous construction of aC �-algebra replacingthe multiplicative semigroup N� by Z�, i.e., omitting the condition of positivity on themultiplicative part of the ax C b-(semi)group. We obtain a purely infinite C �-algebraQZ which can be written as a crossed product of QN by Z=2. The fixed point algebrafor its canonical one-parameter group .�t / is a dihedral group analogue of the Bunce–Deddens algebra. ItsK-theory involves a shift of parity fromK0 toK1 and vice versa.Its stabilization is isomorphic to the natural crossed product C0.Af / Ì PQ.

2 A canonical representation of the ax C b-semigroup

We denote by N the set of natural numbers including 0. N will normally be regardedas a semigroup with addition.

�Research supported by the Deutsche Forschungsgemeinschaft.

202 J. Cuntz

We denote by N� the set of natural numbers excluding 0. N� will normally beregarded as a semigroup with multiplication.

The natural analogue, for a semigroup S , of an unitary representation of a group isa representation of S by isometries, i.e. by operators sg , g 2 S on a Hilbert space thatsatisfy s�gsg D 1.

On the Hilbert space `2.N/ consider the isometries sn, n 2 N� and vk , k 2 Ndefined by

vk.�m/ D �mCk; sn.�m/ D �mnwhere �m; m 2 N denotes the standard orthonormal basis.

We have snsm D snm and vnvm D vnCm, i.e. s and v define representations of N�and N respectively, by isometries. Moreover we have the following relation

snvk D vnksn

which expresses the compatibility between multiplication and addition.In other words, the sn and vk define a representation of the ax C b-semigroup

PN D��1 k

0 n

�j n 2 N�; k 2 N

�

over N, where the matrix�1 k0 n

�is represented by .vksn/�.

Note that, for each n, the operators sn; vsn; : : : ; vn�1sn generate a C �-algebraisomorphic to On, [4].

3 A purely infinite simple C �-algebra associated with theax C b-semigroup

The C �-algebraA generated by the elements sn and v considered in Section 2 containsthe algebra K of compact operators on `2.N/. Denote by u, resp. uk the image of v,resp. vk in the quotientA=K . We also still denote by sn the image of sn in the quotient.Then the uk are unitary and are defined also for k 2 Z and they furthermore satisfythe characteristic relation

Pn�1kD0 ukenu�k D 1 where en D sns

�n denotes the range

projection of sn. This relation expresses the fact that N is the union of the sets ofnumbers which are congruent to k mod n for k D 0; : : : ; n � 1.

We now consider the universal C �-algebra generated by elements satisfying theserelations.

Definition 3.1. We define theC �-algebra QN as the universalC �-algebra generated byisometries sn, n 2 N� with range projections en D sns�n , and by a unitary u satisfyingthe relations

snsm D snm; snu D unsn;n�1XkD0

ukenu�k D 1

for n;m 2 N�.

C�-algebras associated with the ax C b-semigroup over N 203

Lemma 3.2. In QN we have

(a) en DPm�1iD0 uinenmu�in for all n;m 2 N�;

(b) epsq D epqsp D sqep and epeq D epq D eqep when p and q are relativelyprime;

(c) s�nsm D sms�n for n, m relatively prime.

Proof. (a) This follows by conjugating the identity 1 D Puiemu

�i by sn t s�n andusing the fact that snems�n D enm.

(b) Since the uiepqu�i are pairwise orthogonal for 0 � i < pq, we see thatulpepqu

�lp ? ukqepqu�kq if 0 < l < q; 0 < k < p. Thus, using (a),

epsq Dq�1XlD0

ulpepqu�lp

p�1XkD0

ukqepqu�kqsq D epqsq :

This obviously implies epeq D epq and, by symmetry eqep D epq . In particular, epand eq commute.

(c) Using (b) we get

sp.s�p sq/s

�q D epeq D epq D spqs�pq D sp.sqs�p /s�q :

Since sp; sq are isometries this implies that the expressions in parentheses on the leftand right hand side are equal. Here we have, in a first step, assumed that p; q are prime.However any sn is a product of sp’s with p prime.

From Lemma 3.2 (a) it follows that any two of the projections uienu�i commute.We denote byD the commutative subalgebra of QN generated by all these projections.

We also denote by F the subalgebra of QN generated by u and the projections en,n 2 N�.

To analyze the structure of QN further we write it as an inductive limit of thesubalgebrasBn generated by sp1 ; sp2 ; : : : ; spn andu, wherep1; p2; : : : ; pn denote the nfirst prime numbers. Each Bn contains a natural (maximal) commutative subalgebraDn generated by all projections of the form ukemu

�k where m is a product of powersof the p1; : : : ; pn (i.e. a natural number that contains only the pi as prime factors).

Lemma 3.3. The spectrum SpecDn ofDn can be identified canonically with the com-pact space

f0; : : : ; p1 � 1gN � � � � � f0; : : : ; pn � 1gN Š OZp1 � � � � � OZpnProof. Dn is the inductive limit of the subalgebrasD.k/

n Š Clk with lk D pk1pk2 : : : pkn .

The algebra D.k/n is generated by the pairwise orthogonal projections uielku

�i ; 0 �i < lk and in fact, by Lemma 3.2 (a), D.k/

n is the k-fold tensor product of D.1/n by

itself.

204 J. Cuntz

Consider the action of Tn on Bn given by

˛.t1;:::;tn/.spi / D tispiand denote by Fn the fixed-point algebra for ˛ (i.e. Fn D F \Bn). There is a naturalfaithful conditional expectation E W Bn ! Fn defined by E.x/ D R

Tn ˛t .x/dt .Now Dn is the fixed point algebra, in Fn, for the action ˇ of T on Fn given by

ˇt .ek/ D ek; ˇt .u/ D eituand there is an associated expectation F W Fn ! Dn defined by F.x/ D R

T ˇt .x/dt .The composition G D F ı E gives a faithful conditional expectation An ! Dn.These conditional expectations extend to the inductive limit and thus give conditionalexpectations E W QN ! F , F W F ! D and G W QN ! D .

Theorem 3.4. The C �-algebra QN is simple and purely infinite.

Proof. Since inductive limits of purely infinite simple C �-algebras are purely infinitesimple [8], 4.1.8 (ii), it suffices to show that each Bn is purely infinite simple.

For each N , denote, as above, by D.N/n the subalgebra of Dn generated by

fukelu�k; k 2 Zgwhere l D pN1 pN2 : : : pNn . The natural map SpecDn ! SpecD.N/n

is surjective and, by the proof of Lemma 3.3, D.N/n Š Cl .

Choose �1; �2; : : : ; �l in SpecDn such that f�1; �2; : : : ; �lg !SpecD.N/n is bijective

and such that�i .sk t s�k / ¤ �i .t/

for allk of the formk D pm11 : : : pmnn withmi � N . We can choose pairwise orthogonal

projections f1; f2; : : : ; fl inDn with sufficiently small support around the �i such thatfiskfi D 0 for all 1 � i � l and k as above and such that fixfi D �i .x/fi for allx 2 DN

n . Then the map ' W D.N/n ! C �.f1; : : : ; fl/ Š Cl defined by x 7!P

fixfiis an isomorphism.

We denote by P .N/ the set of linear combinations of all products of the formuks

m1p1 : : : s

mnpn with mi � N . For x 2 P .N/ we have that

'.G.x// DX

fixfi :

Let now 0 � x 2 Bn be different from 0. Since G is faithful, G.x/ ¤ 0 and wenormalize x such that kG.x/k D 1. Let y 2 P .N/, for sufficiently large N , be suchthat kx � yk < " � 1. We may also assume that kG.y/k D 1. Then there exists i0,1 � i0 � l such that fi0yfi0 D fi0 . Moreover, there exists an isometry s 2 Bn (of theform s D uksm) such that s�fi0s D 1.

In conclusion, we have s�ys D 1 and

ks�xs � 1k D ks�xs � s�ysk D ks�.x � y/sk < ":This shows that s�xs is invertible.


As a consequence of the simplicity of QN we see that the canonical representationon `2.Z/ by

sn.�k/ D �nk; un.�k/ D �kCnis faithful. Similarly, if we divide the C �-algebra generated by the analogous isome-tries on `2.N/, discussed in Section 2, by the canonical ideal K , we get an algebraisomorphic to QN .

The subalgebra F is generated by u and the projections en, thus by a weightedshift. Thus we recognize F as the Bunce–Deddens algebra of the type where everyprime appears with infinite multiplicity. This well known algebra has been introducedin [3] in exactly that form. It is simple and has a unique tracial state. It can also berepresented as inductive limit of the inductive system .MnC.S1// with maps

Mn.C.S1// �!Mnk.C.S

1//

mapping the unitary u generating C.S1/ to the k � k-matrix0BBBB@

0 0 : : : u

1 0 : : : 0

0 1 : : : 0

: : :

0 : : : 1 0

1CCCCA :

From this description it immediately follows that K0.F / D Q and K1.F / D Z.

Remark 3.5. (a) Just as On is a crossed product of a UHF -algebra by N, we see thatQN is a crossed product F Ì N� by the multiplicative semigroup N�. The algebraQN also contains the commutative subalgebraD. SinceD is the inductive limit of theDn, we see from Lemma 3.3 that SpecD D OZ WD Q

pOZp . The Bost–Connes algebra

CQ [1] can be described as a crossed product C. OZ/ Ì N� and is a natural subalgebraof QN (using the natural inclusion C. OZ/! F ).

We can also obtain QN by adding to the generators �n; n 2 N (our sn) and e� ,� 2 Q=Z for CQ, described in [1], Proposition 18, one additional unitary generator usatisfying

ue� D �e�u and un�n D �nu(here we identify an element � of Q=Z with the corresponding complex number ofmodulus 1).

(b) The algebra generated by sn (for a single n) and u has also been considered in[5]. It has been shown there that this algebra is simple and contains a Bunce–Deddensalgebra.

4 The canonical action of R on QN

So far, in our discussion, the isometries associated with each prime number appear asgenerators on the same footing and it is a priori not clear how to determine, for two

206 J. Cuntz

prime numbers p and q, the size of p and q (or even the bigger number among p and q)from the corresponding generators sp and sq . In fact, the C �-algebra generated by thesn, n 2 N is the infinite tensor product of one Toeplitz algebra for each prime numberand in this C �-algebra there is no way to distinguish the sp for different p.

However, the fact that we have added u to the generators allows to retrieve the nfrom sn using the KMS-condition.

Definition 4.1. Let .A; �t / be a C �-algebra equipped with a one-parameter automor-phism group .�t /, � a state on A and ˇ 2 .0;1�. We say that � satisfies the ˇ-KMS-condition with respect to .�t /, if for each pair x; y of elements inA, there is a holomor-phic function Fx;y , continuous on the boundary, on the strip fz 2 C j Im z 2 Œ0; ˇ�gsuch that

Fx;y.t/ D �.x�t .y//; Fx;y.t C iˇ/ D �.�t .y/x/for t 2 R.

Proposition 4.2. Let �0 be the unique tracial state on F and define � on QN by� D �0 ı E. Let .�t / denote the one-parameter automorphism group on QN definedby �t .sn/ D nitsn and �t .u/ D u. Then � is a 1-KMS-state for .�t /.

Proof. According to [2] 5.3.1, it suffices to check that

�.x�i .y// D �.yx/for a dense *-subalgebra of analytic vectors for .�t /. Here .�z/ denotes the extensionto complex variables z 2 C of .�t / on the set of analytic vectors.

However it is immediately clear that this identity holds for x; y linear combinationsof elements of the form asn or s�mb, a; b 2 F . Such linear combinations are analyticand dense.

Theorem 4.3. There is a unique state � on QN with the following property.There exists a one-parameter automorphism group .�t /t2R for which � is a

1-KMS-state and such that �t .u/ D u, �t .en/ D en for all n and t . Moreover wehave

• � is given by � D �0 ıE where �0 is the canonical trace on F ,

• the one-parameter group for which � is 1-KMS, is unique and is the standardautomorphism group considered above, determined by

�t .sn/ D nitsn and �t .u/ D u:

Proof. Since�t acts as the identity on en and onu, it is the identity on F D C �.u; feng/.If � is a KMS-state for .�t /, it therefore has to be a trace on F . It is well-known (andclear) that there is a unique trace �0 on F . For instance, the relation

X0�k<n

ukenu�k D 1


shows that �0.en/ D 1=n and �0.u/ D 0.The relation

�.s�nsn/ D n�.sns�n/and the 1-KMS-condition show that �t .sn/ D nitsn.

5 The K -groups of QN

TheK-groups of QN can be computed using the fact that QN is a crossed product of theBunce–Deddens algebra F by the semigroup N�. TheK-groups of a Bunce–Deddensalgebra are well known and easy to determine using its representation as an inductivelimit of algebras of type Mn.C.S

1//. Specifically, for F , we have

K0.F / D Q; K1.F / D Z:

We consider QN as an inductive limit of the subalgebrasBn D C �.F ; sp1 ; : : : ; spn/where 2 D p1 < p2 < � � � is the sequence of prime numbers in natural order. NowBnC1 can be considered as a crossed product of Bn by the action of the semigroup Ngiven by conjugation by spnC1

(in fact BnC1 is Morita equivalent to a crossed productby Z of an algebra Morita equivalent to Bn), just as in [4]. Thus the K-groups of Bncan be determined inductively using the Pimsner–Voiculescu sequence.

Theorem 5.1. The K-groups of Bn are given by

K0.Bn/ Š Z2n�1

; K1.Bn/ Š Z2n�1

:

Proof. The first application of the Pimsner–Voiculescu sequence gives an exact se-quence

� � � �! K1.B1/ �! Qid�˛��! Q �! K0.B1/ �! Z

id�˛��! Z �! K1.B1/ �! � � �

where ˛� is the map induced by Ad s1 on K0; K1.Since ˛� D 2 on K0.F / and ˛� D 1 on K1.F / this gives K0.B1/ D Z,

K1.B1/ D Z. In the following steps ˛i D Ad spi induces 1 on K0 and on K1.Thus each consecutive prime pi doubles the number of generators of K0 and K1.

Remark 5.2. It follows that the groupsK0.QN/ andK1.QN/ are free abelian with in-finitely many generators. By construction, the elements ofK0.QN/ are obtained as Bottprojections from the “commuting” elements u and sp1 ; : : : ; sp2n�1

where p1; : : : p2n�1is any choice of an odd number of different prime numbers. Similarly, the elementsof K1.QN/ are obtained as a K-theory product of such elements in K0 with one addi-tional sp2n . Therefore the generators ofK�.QN/ are labeled by the square free numbersof the form p1p2 : : : pk with p1; : : : ; pk distinct primes.

208 J. Cuntz

6 Representations as crossed products

For eachn, define the endomorphism'n of QN by'n.x/ D snxs�n . Since'n'm D 'nmthis defines an inductive system.

Definition 6.1. We define xQN as the inductive limit of the inductive system .QN ; 'n/.

By construction we have a family �n of natural inclusions of QN into the inductivelimit xQN satisfying the relations �nm'n D �m. We denote by 1k the element �k.1/ ofxQN . We have that 1k D �kl.el/ � 1kl . The union of the subalgebras 1k xQN1k is densein xQN and 1k � 1kl for all k; l . In order to define a multiplier a of xQN it thereforesuffices to define a1k and 1ka for all k.

We can extend the isometries sn naturally to unitaries Nsn in the multiplier algebraM.xQN/ of xQN by requiring

Nsn1k D �k.sn/; 1k Nsn D �kn.sn/:Note that this is well defined because, using the identities snel D enlsn and 'l.sn/ Dsnel , we have

.1k Nsn/1l D �kn.sn/�l.1/D �knl.snelekn/D �knl.enlsnekn/D �k.1/�l.sn/D 1k.Nsn1l/:Proposition 6.2. The elements Nsn, n 2 N, define unitaries in M.xQN/ such that Nsn Nsm DNsnm and such that Nsn Ns�m D Ns�m Nsn.

Defining Nsa D Nsn Ns�m for a D n=m 2 Q�C we define unitaries in M.xQN/ such thatNsa Nsb D Nsab for a; b 2 Q�C.

Proof. In order to show that Nsn is unitary it suffices to show that 1k Nsn Ns�n D 1k and1k Ns�n Nsn D 1k for all k, n.

We can also extend the generating unitary u in QN to a unitary in the multiplieralgebra M.xQN/. We define the unitary Nu in M.xQN/ by the identity

Nu 1k D �k.uk/:We can also define fractional powers of Nu by setting

Nu1=n 1kn D �kn.uk/:Proposition 6.3. For all a 2 Q�C and b 2 Q we have the identity

Nsa Nub D Nuab Nsa:Proof. Check that Ns1=n Nu D Nu1=n Ns1=n

Following Bost–Connes we denote by PCQ the ax C b-group

PCQ D��1 b

0 a

�j a 2 Q�C; b 2 Q

�:


It follows from the previous proposition that we have a representation of PCQ in the

unitary group of M.xQN/. Denote by Af the locally compact space of finite adeles overQ, i.e.,

Af D f.xp/p2P j xp 2 OQp and xp 2 OZp for almost all pgwhere P is the set of primes in N.

The canonical commutative subalgebra D of QN is by the Gelfand transform iso-morphic to C.X/ where X is the compact space OZ D Q

p2POZp . It is invariant under

the endomorphisms 'n and we obtain an inductive system of commutative algebras.D; 'n/. The inductive limit xD of this system is a canonical commutative subalgebraof xQN . It is isomorphic to C0.Af /. In fact the spectrum of xD is the projective limit ofthe system (SpecD; O'n/ and 'n corresponds to multiplication by n on OZ.

Theorem 6.4. The algebra xQN is isomorphic to the crossed product of C0.Af / by thenatural action of the ax C b-group PCQ .

Proof. Denote by B the crossed product and consider the projection e 2 C0.Af / � Bdefined by the characteristic function of the maximal compact subgroup

Qp2P

OZp �Af . Consider also the multipliers Ns�n and Nu� ofB defined as the images of the elements�1 00 n

�and

�1 10 1

�of PCQ , respectively. Then the elements sn D Nsne and u D Nue

satisfy the relations defining QN . Moreover, the C �-algebra generated by the sn and ucontains the subalgebra eC0.Af / of B . This shows that this subalgebra equals eBeand, by simplicity of QN , it follows that eBe Š QN .

It is a somewhat surprising fact that we get exactly the same C �-algebra, eventogether with its canonical action of R if we replace the completion Af of Q at thefinite places by the completion R at the infinite place.

Theorem 6.5. The algebra xQN is isomorphic to the crossed product of C0.R/ by thenatural action of the ax+b-group PCQ . There is an isomorphism xQN ! C0.R/ Ì PCQwhich carries the natural one parameter group �t to a one parameter group �0t suchthat �0t .f ug/ D a�itf ug where ug denotes the unitary multiplier of C0.R/ Ì PCQassociated with an element g 2 PCQ of the form

g D�1 b

0 a

�

(the KMS-state � is carried to the state which extends Lebesgue measure on C0.R/).

In order to prove this we need a little bit of preparation concerning the representationof the Bunce–Deddens algebra F as an inductive limit. As noted above it is an inductivelimit of the inductive system .Mn.C.S

1/// with maps sending the unitary generator zof C.S1/ to a unitary v in Mk.C/˝ C.S1/ satisfying vk D 1˝ z. We observe nowthat such a v is unique up to unitary equivalence.

Lemma 6.6. Let v1; v2 be two unitaries inMk.C/˝C.S1/ such that vk1 D vk2 D 1˝z.Then there is a unitary w in Mk.C/˝ C.S1/ such that v2 D wv1w�.

210 J. Cuntz

Proof. The spectral projections pi .t/ for the different k � th roots of e2�it1, given byv1.t/ and v2.t/, have to be continuous functions of t . This implies that all the pi .t/ areone-dimensional and that, after possibly relabelling, we must have the situation where

pi .t C 1/ D piC1.t/:This means that the pi combine to define a line bundle on the k-fold covering of S1

by S1. However any two such line bundles are unitarily equivalent.

We now determine the crossed product C0.R/ Ì Q where Q acts by translation.

Lemma 6.7. The crossed product C0.R/ Ì Q is isomorphic to the stabilized Bunce–Deddens algebra K ˝ F .

Proof. The algebra C0.R/Ì Q is an inductive limit of algebras of the form C0.R/Ì Zwith respect to the maps

ˇk W C0.R/ Ì Z �! C0.R/ Ì Z

obtained from the embeddings Z Š Z 1k,! Q. It is well known that C0.R/ Ì Z is

isomorphic to K ˝ C.S1/. An explicit isomorphism is obtained from the map

Xn2Z

fnun 7�!

Xn2Z;k2Z

�k.fn/ek;kCn

where �k denotes translation by k, uk denotes the unitary in the crossed product im-plementing this automorphism and eij denote the matrix units in K ŠK.`2Z/.

This map sends C0.R/ Ì Z to the mapping torus algebra

ff 2 C.R;K/ j f .t C 1/ D Uf .t/U �g ŠK ˝ C.S1/

where U is the multiplier of K DK.`2.Z// given by the bilateral shift on `2.Z/.A projection p corresponding, under this isomorphism, to e ˝ 1 in K ˝ C.S1/

with e a projection of rank 1 can be represented in the form p D ug C f C gu�with appropriate positive functions f and g with compact support on R. Underthe map C0.R/ Ì Z Š C0.R/ Ì kZ ! C0.R/ Ì Z, the projection p is mapped top0 D ukgk C fk C gku

�k where gk.t/ WD g.t=k/, fk.t/ WD f .t=k/. Now, p0corresponds to a projection of rank k in K ˝ C.S1/.

Let z be the unitary generator of C.S1/. Then the element e ˝ z corresponds thefunction e2�it .ugCf Cgu�/which is mapped to v D e2�it=k.ukgkCfkCgku�k/.Thus vk D p0 and p0 corresponds to a projection of rank k in K ˝ C.S1/.

On the other hand F D lim�! An where An D Mn.C.S1// and the inductive limit

is taken relative to the maps ˛k W Mn.C.S1//! Mkn.C.S

1// which map the unitarygenerator z of C.S1/ to an element v such that vk D 1˝ z. Compare this now to theinductive system A0n D C0.R/ Ì Z 1

nwith respect to the maps ˇk considered above.


From our analysis of ˇk and from Lemma 6.6 we conclude that there are unitaries Wkin M.A0

kn/ such that the following diagram commutes

An

��

˛k �� Akn

��

A0nAdWkˇk �� A0

kn

where the vertical arrows denote the natural inclusions Mn.C.S1//!K ˝ C.S1/.

We conclude that these natural inclusions induce an isomorphism from K ˝ F Dlim�!K ˝ An to C0.R/ Ì Q D lim�! A

0n.

Proof of Theorem 6.5. Consider the commutative diagram and the injection F ,!C0.R/ Ì Q constructed at the end of the proof of Lemma 6.7. Note also that theˇk are -unital and therefore extend to the multiplier algebra. This shows that theinjection transforms p into p times an approximately inner automorphism, for eachprime p. Therefore the injection transforms �t , which is the dual action on the crossedproduct for the character n 7! nit of N�, to the restriction of the corresponding dualaction on the crossed product .C0.R/ Ì Q/ Ì N�.

Finally, note that the endomorphisms ˇn of C0.R/ Ì Q are in fact automorphismsand that therefore ˇ extends from a semigroup action to an action of the group Q�C.This shows that .C0.R/ Ì Q/ Ì N� D .C0.R/ Ì Q/ Ì Q�C D C0.R/ Ì PCQ .

Remark 6.8. If we consider the full space of adeles QA D Af �R rather than the finiteadeles Af or the completion at the infinite place R, we obtain the following situation.By [9], IV, §2, Lemma 2, Q is discrete in QA and the quotient is the compact space. OZ � R/=Z. Thus, the crossed product C0.QA/ Ì Q which would be analogous tothe Bunce–Deddens algebra is Morita equivalent to C.. OZ �R/=Z/. Now, . OZ �R/=Zhas a measure which is invariant under the action of the multiplicative semigroup N�.Therefore the crossed product C0.QA/ÌPCQ has a trace and is not isomorphic to QN .

7 The case of the multiplicative semigroup Z�

We consider now the analogousC �-algebras where we replace the multiplicative semi-group N� by the semigroup Z�. This case is also important in view of generalizationsfrom Q to more general number fields (in fact the construction of QZ and much of theanalysis of its structure carries over to the ring of integers in an arbitrary algebraic num-ber field, at least if the class number is 1). Thus, on `2.Z/ we consider the isometriessn; n 2 Z� and the unitaries um; m 2 Z defined by sn.�k/ D �nk and um.�k/ D �kCm.As above, these operators satisfy the relations

snsm D snm; snum D unmsn;

n�1XiD0

uienu�i D 1: (1)

212 J. Cuntz

The operator s�1 plays a somewhat special role and we therefore denote it by f . Thenthe sn; n 2 Z and u generate the same C �-algebra as the sn; n 2 N together with u andf . The element f is a selfadjoint unitary so that f 2 D 1 and we have the relations

f sn D snf; f uf D u�1:We consider now again the universal C �-algebra generated by isometries sn; n 2 Zand a unitary u subject to the relations (1). We denote this C �-algebra by QZ. We seefrom the discussion above that we get a crossed product QZ Š QN Ì Z=2 where Z=2acts by the automorphism ˛ of QN that fixes the sn; n 2 N and ˛.u/ D u�1.

Theorem 7.1. The algebra QZ is simple and purely infinite.

Proof. Composing the conditional expectation G W QN ! D used in the proof ofTheorem 3.4 with the natural expectation QZ D QN Ì Z=2! QN we obtain again afaithful expectation G0 W QZ ! D . The rest of the proof follows exactly the proof ofTheorem 3.4, using in addition the fact that the fi in that proof can be chosen such thatfiffi D 0.

Denote by PQ the full ax C b-group over Q, i.e.

PQ D��1 b

0 a

�j a 2 Q�; b 2 Q

�:

Theorem 7.2. QZ is isomorphic to the crossed product of C0.Af / or of C0.R/ by thenatural action of PQ.

Proof. This follows from Theorems 6.4 and 6.5 since PQ D PCQ Ì Z=2.

On QZ we can define the one-parameter group .�t / by �t .sn/ D nitsn; n 2 N�.The fixed point algebra is the crossed product F Ì Z=2 of the Bunce–Deddens algebraby Z=2.

In order to compute theK-groups of QZ we first determine theK-theory for F 0 DF Ì Z=2. This algebra is the inductive limit of the subalgebras A0n D C �.u; f; en/.Lemma 7.3. (a) The C �-algebra C �.u; f / is isomorphic to C �.D/, where D is thedihedral group D D Z Ì Z=2 (Z=2 acts on Z by a 7! �a). For each n D 1; 2; : : : ,the algebra A0n is isomorphic to Mn.C

�.D//.(b) We have K0.A0n/ D K0.C

�.D// D Z3 and K1.A0n/ D K1.C�.D// D 0 for

all n. The generators ofK0.C �.D// are given by Œ1� and by the classes of the spectralprojections .uf /C and f C of uf and f , for the eigenvalue 1.

(c) Let p be prime. If p D 2, then the mapK0.A0n/! K0.A0pn/ is described, with

respect to the basis Œ1�, Œ.uf /C� and Œf C� of K0.C �.D// Š Z3, by the matrix0@2 1 0

0 0 1

0 0 1

1A


If p is odd, then the map K0.A0n/! K0.A0pn/ is described by the matrix

0@p

p�12

p�12

0 0 1

0 1 0

1A :

Proof. (a) It is clear that the universal algebra generated by two unitariesu; f satisfyingf 2 D 1 and f uf D u� is isomorphic to C �.D/.

In the decomposition of C �.u; f; e2/ with respect to the orthogonal projections e2and ue2u�1, the elements u and f correspond to matrices�

0 w

1 0

�and

�f0 0

0 f1

�

where w is unitary and f0; f1 are symmetries (selfadjoint unitaries).The relations between u and f imply that wf1 D f0 and that wf1 is a selfadjoint

unitary, whence f1wf1 D w�. Thus A02 is isomorphic to M2.C�.w; f1// and w; f1

satisfy the same relations as u; f .If p is an odd prime, then in the decomposition of C �.u; f; ep/ with respect to the

pairwise orthogonal projections ep; uepu�1; : : : ; up�1epu�.p�1/, u and f correspondto the following p � p-matrices

0BBBB@

0 0 : : : w

1 0 : : : 0

0 1 : : : 0

: : :

0 : : : 1 0

1CCCCA and

0BBBBBB@

f0 0 : : : 0

0 0 : : : f �10 0 : : : f �2 0

: : :

0 0 f2 : : :

0 f1 : : : 0 0

1CCCCCCA

where w and the f1; : : : ; fp�12

are unitary and f0 is a symmetry.

The relation f uf D u� implies that f0 D wf1, f1 D f2 D � � � D fp�12

and f 2i D1 for all i . From .wf1/

2 D 1we derive that f1wf1 D w�. ThusA0p ŠMp.C �.w; f1//and w; f1 satisfy the same relations as u; f .

The case of general n is obtained by iteration using the fact that C �.u; f; enm/ ŠMm.C

�.u; f; en//.(b) This follows for instance from the well known fact that C �.D/Š .C ?C/�.(c) Letp D 2. Using the description ofK0.C �.u; f // under (b) and the description

of the map C �.u; f /! C �.u; f; e2/ from the proof of (a), we see that the generatorsŒ1�, Œ.uf /C� and Œf C� are respectively mapped to 2Œ1�, Œ1� and Œf C�C Œ.uf /C�.

Letp be an odd prime. The matrix corresponding tof in the proof of (a) is conjugateto the matrix where all the fi , i D 1; : : : ; .p � 1/=2 are replaced by 1. Thus the classof f C is mapped to Œ.wf1/C�C p�1

2Œ1�.

The matrix corresponding to uf is conjugate to a selfadjoint matrix of the sameform as the matrix representing the image of f , but with the upper left entry f0 replacedby f1. Thus the class of .uf /C is mapped to Œf C1 �C p�1

2Œ1�.

214 J. Cuntz

Proposition 7.4. TheK-groups of the algebra F 0 D F ÌZ=2 are given byK0.F 0/ DQ˚ Z and K1.F 0/ D 0.

Proof. This follows from the fact that F 0 is the inductive limit of the algebras A0n andthe description of the mapsK0.A0n/! K0.A

0pn/ given in Lemma 7.3 (c). The element

ŒfC� C ŒufC� � Œ1� is invariant under the map K0.A0n/ ! K0.A0pn/ and maps to the

generator of Z in the inductive limit.

Remark 7.5. Since F 0 is a simple C �-algebra with unique trace in a classifiableclass (in the sense of the classification program), this computation of its K-theory incombination with the classification result in [6] shows that F 0 is an AF -algebra. Thuswe obtain an example of a simple AF -algebra F 0 admitting an action by Z=2 suchthat the fixed point algebra is the Bunce–Deddens algebra F and thus not AF . Thisexample had been considered before in [7].

In analogy to the case over N we denote byB 0n the C �-subalgebra of QZ generatedby u; f; sp1 ; : : : ; spn . We now immediately deduce

Theorem 7.6. We have

K0.B0n/ D Z2

n�1

; K1.B0n/ D Z2

n�1

and

K0.QZ/ D Z1; K1.QZ/ D Z1:

Note added in proof. The arguments in the computation of the K-theory are notcomplete. This problem will be addressed in a subsequent paper.

References

[1] J.-B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spon-taneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), 411–457.

[2] Ola Bratteli and Derek W. Robinson, Operator algebras and quantum statistical mechanics,Vol. 2, Equilibrium states. Models in quantum statistical mechanics, sec. ed., Texts Monogr.Phys., Springer-Verlag, Berlin 1997.

[3] John W. Bunce and James A. Deddens, A family of simple C�-algebras related to weightedshift operators, J. Functional Analysis 19(1975), 13–24.

[4] Joachim Cuntz, SimpleC�-algebras generated by isometries, Comm. Math. Phys. 57 (1977),173–185.

[5] Ilan Hirshberg, On C�-algebras associated to certain endomorphisms of discrete groups,New York J. Math. 8 (2002), 99–109.

[6] Xinhui Jiang and Hongbing Su, A classification of simple limits of splitting interval algebras,J. Funct. Anal. 151 (1997), 50–76.

[7] A. Kumjian, An involutive automorphism of the Bunce-Deddens algebra, C. R. Math. Rep.Acad. Sci. Canada 10 (1988), 217–218.


[8] M. Rørdam, Classification of nuclear, simple C�-algebras, in Classification of nuclearC�-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci. 126, Springer-Verlag,Berlin 2002, 1–145.

[9] André Weil, Basic number theory, Grundlehren Math. Wiss. 144, Springer-Verlag, NewYork1967.

On a class of Hilbert C*-manifolds

Wend Werner

1 Introduction

This note had initially two objectives, an explicit calculation, for all vector fields, ofthe invariant connection on a certain type of infinite dimensional symmetric space and,using results from [1], to characterize those invariant cone fields on a similar kindof spaces that can be thought of as the result of some kind of ‘quantization’. Bothquestions are related since the invariance of the cone fields is intimately connected tothe behavior of parallel transport along geodesics.

Invariant connections for finite dimensional symmetric spaces have long been knownto exist and to be unique. One has to be a little bit more careful in the infinite dimen-sional, Banach manifold setting since there the existence of a sufficient amount ofsmooth functions no longer can be proven. The type of calculations we are interestedin here have been carried out under similar circumstances in [2].

Important for our approach is to use an invariant Hilbert C*-structure on the fibersof the tangent bundle. We show that the symmetric space we are dealing with canbe defined in terms of the automorphism group of this structure. For the underlyinginvariant (operator) Finsler structure, the analogous result holds.

The theory of invariant cone fields on finite dimensional spaces is very well devel-oped. A comprehensive account is [5]. We will see in the last section that the infinitedimensional theory might behave slightly differently.

2 Hilbert C*-manifolds

2.1. Recall that a (left) Hilbert C*-module over a C*-algebra A is a complex vectorspace E which is a left A-module with a sesquilinear pairing E � E ! A satisfying,for r; s 2 E and a 2 A, the following requirements:

(i) har; si D ahr; si;(ii) hr; si D hs; ri�;

(iii) hs; si > 0 for s ¤ 0;

(iv) equipped with the normksk D

pkhs; sik;

E is a Banach space.

Right Hilbert C*-modules are defined similarly. Whenever we want to refer to thealgebra A explicitly, we speak of a Hilbert A-module.

218 W. Werner

2.2. The objects defined above coincide with the so called ternary rings of operators(TRO), which are intrinsically characterized in [9], [11]. On such a space E, a tripleproduct f � ; � ; � g is given in such a way that E, up to (the obvious definition of) TRO-isomorphisms, is a subspace of a space of bounded Hilbert space operators L.H/,invariant under the triple product

fx; y; zg D xy�z:The relation to (left) Hilbert C*-modules is based on the equation

fx; y; zg D hx; yiz;connecting triple product to module action as well as scalar product of a HilbertA-module. Here, A is the algebra of linear mappings

A D EE� D lin fx 7! fe1; e2; xg j e1;2 2 Eg ;which is independent of the chosen embedding. Note that the norm of an element e 2 Emust coincide with kek D kfe; e; egk1=3.

2.3. TRO-morphisms will be those mappings that preserve the product f � ; � ; � g. Thesemappings differ in general from what is considered to be the natural choice for HilbertA-morphisms, the so called adjointable maps. The latter are in particular A-modulemorphisms, a property that would be too restrictive for our purposes. We will hencestick in the following to TRO-morphisms.

Definition 2.4. Let M be a Banach manifold and A a C*-algebra. M is said to bea (right-, left-) Hilbert A-manifold if on each tangent space Tp.M/ there is given thestructure of a Hilbert A-module depending smoothly on base points.

Definition 2.5. Let M be a Hilbert C*-manifold. The group of automorphisms,AutM consists of all diffeomorphisms ˆ W M ! M so that dˆ is (pointwise) aTRO-morphism.

2.6. It is not clear under which circumstances a Banach manifold can be given thestructure of a Hilbert C*-manifold. This is so because, first, no characterization seemsto be known of Banach spaces that are (topologically linear) isomorphic to HilbertC*-modules, and, second, because in general, there is no smooth partition of the unitthat would permit the step from local to global.

We will use group actions instead. This will bring up homogeneous Hilbert C*-manifolds in the following sense.

Definition. A Banach manifoldM is called homogeneous Hilbert C*-manifold iff it isa Hilbert C*-manifold for which AutM acts transitively.

2.7. Suppose M is a homogeneous space with respect to a smooth Banach Lie groupaction. Fix a base point o 2M , and denote the isotropy subgroup at o by H . Supposethat To.M/ carries the structure of a Hilbert A-module with form h � ; � io, module

On a class of Hilbert C*-manifolds 219

map x 7! a �o x, and TRO-structure fx; y; zgo D hx; yio �o z. If for all h 2 H ,x; y; z 2 To.M/ we have

dohfx; y; zgo D fdoh.x/; doh.y/; doh.z/gothen a Hilbert A-module on Tp.M/ for any p D g.o/ 2M , is well-defined through

fx; y; zgp D dogfdpg�1.x/; dpg�1.y/; dpg�1.z/go:

With this structure, M becomes a homogeneous Hilbert C*-manifold

2.8. Our example here is the following. Fix a TRO E and denote by U its open unitball. If we follow the path laid out above, we find the following invariant HilbertC*-structure on U . Define a triple product for TaM at a 2 U by

fx; y; zga D x.1 � a�a/�1y�.1 � aa�/�1z;

so thathx; yia D .1 � aa�/�1=2x.1 � a�a/�1y�.1 � aa�/�1=2

as well as� �a z D .1 � aa�/1=2�.1 � aa�/�1=2; � 2 EE�:

We will refer to this structure as the canonical Hilbert C*-structure on U .

2.9. This definition is motivated in the following way. As shown in [4], HolU , thegroup of all biholomorphic automorphismsU , consists of mappings of the formT ıMa,where for any a 2 U ,

Ma.x/ WD .1 � aa�/�1=2.x C a/.1C a�x/�1.1 � a�a/1=2;

and T is a (linear) isometry of E, restricted to U . Then HolU acts transitively on U ,and the group of (linear) isometries is the isotropy subgroup at the point 0. For lateruse, we include here the fact that

dxMa.h/ D .1 � aa�/1=2.1C xa�/�1h.1C a�x/�1.1 � a�a/1=2

as well as

d2xMa.h1; h2/

D �.1 � aa�/1=2.1C xa�/�1h2a�.1C xa�/h1.1C a�x/�1.1 � a�a/1=2� .1 � aa�/1=2.1C xa�/�1h1.1C a�x/�1a�h2.1C a�x/�1.1 � a�a/1=2:

The Hilbert C*-structure from 2.8 is in fact constructed according to the constructionin 2.6 for a group G, somewhat smaller than HolU .

220 W. Werner

Definition 2.10. Suppose X is a (closed) subspace of L.H/. Equip

Mn.X/ D˚.xij / j xij 2 X for i; j D 1; : : : ; n�

with the norm it carries as a subspace of Mn.L.H// D L.H ˚ � � � ˚H/, and, for abounded operator T W X ! X , denote by T .n/ D idMn.C/ ˝ X W Mn.X/ ! Mn.X/

the operator .xij / 7�! .T xij /. Then T is said to be completely bounded, iff

kT kcb WD supn2NkT .n/k <1:

Similarly, T is called a complete isometry iff each of the maps T .n/ is an isometry.

We have the following

Theorem 2.11 ([3], [9]). Let E be a TRO. Then Mn.E/ carries a distinguished TRO-structure, and the group of TRO-automorphisms of E coincides with the group ofcomplete isometries.

Theorem 2.12. Let U be the open unit ball of a TRO E, equipped with the canoni-cal Hilbert C*-structure, and suppose Mn.E/ carries the standard TRO-structure foreach n 2 N. Then a diffeomorphism ˆ W U ! U is a Hilbert C*-automorphism iffˆ D T ıMa, where a 2 U , and T is the restriction of a linear and completely isometricmapping of E to U .

Proof. That each map of the form T ıMa is in AutU follows from the constructionof the Hilbert C*-structure on U as well as from Theorem 2.11. Since the derivativeof any elementˆ 2 AutU is complex linear by definition, AutU � HolU , and henceˆ D T ıMa for some a 2 U and an isometry T . Because the linear map T fixes theorigin, dT must be a TRO-automorphism, and the result follows, by another applicationof Theorem 2.11.

2.13. Let M again be a Hilbert C*-manifold. Then the tangent space at each pointm of M carries an essentially unique norm, naturally connected to the TRO-structureby kxkm D kfx; x; xgmk1=3. Since TROs embed completely isometrically into somespace of bounded Hilbert space operators, this norm naturally extends to the spacesMn.TmM/. We will call a Banach manifold M an operator Finsler manifold in caseeach tangent space carries the structure of an operator space depending continuously onbase points. In this sense, any Hilbert C*-manifold carries an operator Finsler structurein a natural way. If, as before, U is the open unit ball of a fixed TRO .E; f � ; � ; � g; k � k/,furnished with its invariant Hilbert C*-structure then, using that complete isometriesand automorphisms of a TRO are the same mappings, we have

Theorem. If the open unit ball of a TRO is equipped with the natural invariant operatorFinsler structure its automorphism groups coincides with the automorphism group ofthe underlying homogeneous Hilbert C*-manifold.


3 The invariant connection

3.1. The definition of a connection for Banach manifolds cannot be, due to the scarcityof smooth functions, the usual one. We follow [6], 1.5.1 Definition.

Definition 3.2. Let M be a manifold, modeled over the Banach space E, and denotethe space of bounded bilinear mappings E � E ! E by L2.E;E/. Then M is saidto possess a connection iff there is an atlas U for M so that for each U 2 U there is asmooth mapping � W U ! L2.E;E/, called the Christoffel symbol of the connectionon U , which under a change of coordinates ˆ transforms according to

�.ˆ0X;ˆ0Y / D ˆ00.X; Y /Cˆ0�.X; Y /:The covariant derivative of a vector field Y in the direction of the vector field X is,locally, defined to be the principal part of

rXY D dX.Y / � �.X; Y /;where, in a chart, the principal part of .u;X/ 2 U � T .U / is X .

The reader should note that this definition is equivalent to specifying a smoothvector subbundle H of T TM with the property that Hp is for each p 2 TM closedand complementary to the tangent space ker.dp�/ of the fiber E�.p/ through p. It isnot, however, equivalent to the requirement thatr beC1.M/-linear in its first variableand a derivation w.r.t. the action of C1.M/ on vector fields, although connections asdefined above do have this property.

3.3. We can keep the notion of invariance of a connection under the smooth action ofa (Banach) Lie group, however. In fact, if such a group G acts on M then, for eachg 2 G a connection g�r is defined by letting

g�rXY D rg�Xg�Y; g�X.gm/ D dmgX.m/:

Christoffel symbols then transform as in the definition above,

�g.m/.g0.m/X.m/; g0.m/Y.m// D g00.m/.X.m/; Y.m//C g0.m/�m.X.m/; Y.m//;

and we call r invariant under the action of G whenever g�r D r for all g 2 G.

3.4. If M is the Hilbert C*-manifold U that was defined in the last section, therecan be at most one connection which is invariant under the action of the group ofbiholomorphic self-maps of U . This is because the difference of two of them is thedifference of their Christoffel symbols which have to vanish at each point according tothe way they transform under the reflections �a D Ma�oM�a, �o.x/ D �x. To findone, we let

�o.x; y/ D 0:Then, for any g leaving the origin fixed, g00 D 0 and so �0 remains zero when trans-formed under g. Using the transformation rule for Christoffel symbols we find

222 W. Werner

Theorem 3.5. On the Hilbert C*-manifold U there exists exactly one invariant con-nection whose Christoffel symbol at a is given by

�a.x; y/ D y.1 � a�a/�1=2a�.1 � aa�/�1=2xC x.1 � a�a/�1=2a�.1 � aa�/�1=2y D 2fx;M 0a.0/a; ygsa;

where fx; y; zgs D 12.fz; y; xg C fx; y; zg/ denotes the (symmetric) Jordan triple

product on E.

3.6. The reader should observe that the form f � ; � ; � g defined above is parallel for theinvariant connection, i.e. for all vector fields X; Y;Z and W we have

rW fX; Y;Zg D frWXYZg C fXrW YZg C fXYrWZg:

This follows either from direct calculation (using the Jacobi identity for f � ; � ; � g), orfrom the fact that the covariant derivative of f � ; � ; � g is invariant under reflections. Thiscondition shows that r behaves like the Levi-Civitá connection with respect to theHilbert C*-structure on M . Under what conditions such a connection exists undermore general circumstances, is investigated in [10].

4 Causality

4.1. It is customary in a number of physical theories to pass from a Lorentzian to aRiemannian manifold (‘Wick Rotation’). This is due to the difficulties one is facedwith in a truly Lorentzian situation and which disappear in the Riemannian set-up. Onepeculiarity of Lorentzian manifolds can still be modeled in the Riemannian situation:Light cones in the fibers of the tangent bundle which specify those pairs of points onMthat may interact with each other and are thus intimately connected to the notion ofcausality.

Definition 4.2. A cone field on a manifold M is, for each m 2M , an assignment of acone Cm � Tm.M/.

Here a cone is always supposed to be pointed but not necessarily to be generating.

4.3. Our interest here is with cone fields that are invariant under the action of a groupG,i.e. that Cgm D dmg.Cm/ for all m 2 M and g 2 G. In our case, G will be a certainsubgroup of AutU . Though it might be tempting to see invariance under HolU (ora large subgroup) as a substitute for diffeomorphism invariance in the complex case,the main motivation here comes from the property that the connection defined in theprevious section has the property that for an invariant cone field .Cm/ parallel transportalong a curve � from �.a/ to �.b/ ends inC�.b/ if it began inC�.a/. This is well knownin the finite dimensional situation (see [7], [8]) and, in this regard, not much is changingin passing to infinite dimensions.


4.4. In the following we will suppose that ‘space’ is represented by a certain set D ofself-adjoint operators, and that on this set there is defined an invariant cone field. Thequestion we would like to answer is this: How can such fields be characterized, whichcome from interpreting the points of D as bounded Hilbert space operators?

In the following, denote by E a fixed (abstract) ternary ring of operators, and by Uits unit ball.

4.5. On top of the causal structure of U we need ‘selfadjointness’, which for us will bethe existence of a ‘real form’ for U , compatible with the (almost) complex structure.We do this by requiring that E carries an involutory real automorphism ‘*’ so that.ix/� D �ix� for all x 2 E. Since we will be studying TRO-embeddings into L.H/that preserve the real form, it will be necessary to impose the additional condition that,for all x; y; z 2 E, fx; y; zg� D fz�; y�; x�g. (Note that then each x 2 E has a uniquedecomposition into real and imaginary parts, with some norm estimates.) A ternaryring of operators E that meets all these conditions, will be called a *-ternary ring ofoperators. We will suppose in the following that E is a space of this kind.

4.6. The ‘space manifold’here will be the open unit ballUsa of the selfadjoint part ofE.Usa is itself a symmetric space. If Gsa consists of all elements in AutU which leaveUsa invariant, then Usa D Gsa=Hsa, whereHsa comprises the TRO-automorphisms thatare *-selfadjoint. In fact, it follows from fx; y; zg� D fz�; y�; x�g for all x; y; z 2 E(and an expansion into power series) that Ma.x/

� D Ma�.x�/ for all x 2 U and soMa 2 Gsa iff a� D a. A TRO-automorphism T is in Hsa iff T .x�/� D T .x/ for allx 2 U , and so

Gsa D fT ıMa j T 2 Hsa; a 2 Usag ;as well as Usa D Gsa=Hsa.

4.7. In order to comply with the requirement that causality be invariant under paralleltransport we have to impose the condition that the field of cones we fix in T Usa mustbe invariant under the action ofGsa. We consider smooth embeddingsˆ W U ! L.H/

which preserve

• the Hilbert C*-structure,

• the complex structure as well as the (canonical) real forms,

• the action of the automorphism groups.

And we want to know: What characterizes the Gsa invariant cone fields that are pulledback to U via ˆ? Whenever a cone field meets these properties we will call it natural.

4.8. Since a natural cone field is supposed to be invariant under the action of Gsa,we may restrict our attention to cones in ToU D E. Furthermore, any cone in Ethat gives rise to a Gsa invariant field of cones has to be invariant under the action ofHsa. It can also be shown that under the assumptions made above, dˆ has to preservethe ternary structure of each tangent space TpU . The question we were asking thusbecomes: What properties must anHsa-invariant cone in Esa possess so that it is of the

224 W. Werner

form ‰�1.L.H/C/ for a *-ternary monomorphism ‰ W E ! L.H/? For the sake ofsimplicity, we restrict our attention to the case where E is a dual Banach space. Then

Theorem 4.9 ([1]). Let E be a *-ternary ring of operators, which is a dual Banachspace. Define the center of E by

Z.E/ D fe 2 E j exy D xye for all x; y 2 Egand call u 2 E tripotent whenever fu; u; ug D u. Then a cone C � Esa is natural iffthere is a central, selfadjoint tripotent element u 2 E so that

C D Cu WD feue� j e 2 Eg :Definition 4.10. If E is a TRO with real form *, and u 2 Esa, then u is called rigid iffˆ.u/ D u for all *-selfadjoint TRO-automorphisms ˆ of E.

The following result is now easy to prove.

Theorem 4.11. The only conesCu inE that give rise to aGsa-invariant causal structureon Usa, coming from an embedding of E into some space L.H/, are those for whichthe central, selfadjoint element u is rigid.

Note that the existence of a rigid selfadjoint central tripotent u is impossible in finitedimensions. In fact, Z.E/ is a weak*-closed commutative von Neumann algebra (seeLemma 4.10 in [1]) in which the tripotents are projections, which is invariant under anyTRO-automorphismˆ ofE, and for whichˆjZ.E/ is a C*-automorphism whose prop-erties essential here are best understood by means of the underlying homeomorphismof the spectrum of Z.E/.

References

[1] D. Blecher and W. Werner, Ordered C�-modules, Proc. London Math. Soc. 92 (2006),682–712.

[2] G. Corach, H. Porta, and L. Recht, The geometry of spaces of projections in C*-algebras,Adv. Math. 101 (1993), 59–77.

[3] M. Hamana, Triple envelopes and Shilov boundaries of operator spaces, Math. J. ToyamaUniv. 22 (1999), 77–93.

[4] L.A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces, inInfinite Dimensional Holomorphy (T. L. Hayden and T. J. Su Suffridge, eds.), Lecture Notesin Math. 364, Springer-Verlag, Berlin 1974, 13–40.

[5] J. Hilgert and G. Olafsson, Causal symmetric spaces, Perspect. Math. 18, Academic Press,San Diego, CA, 1997.

[6] W. Klingenberg, Riemannian Geometry, second edition, de Gruyter Stud. Math. 1, Walterde Gruyter, Berlin 1995.

[7] O. Loos, Symmetric Spaces I, Benjamin, New York, Amsterdam 1969.


[8] D. Mittenhuber and K.-H. Neeb, On the exponential function of an ordered manifold withaffine connection, Math. Z. 218 (1995), 1–23.

[9] M. Neal and B. Russo, Operator space characterizations of C*-algebras and ternary rings,Pacific J. Math. 209 (2003), 339–364.

[10] W. Werner, Hilbert C*-manifolds with Levi-Civitá connection, in preparation.

[11] H. Zettl, A characterization of ternary rings of operators, Adv. in Math. 48 (1983), 117–143.

Duality for topological abelian group stacksand T -duality

Ulrich Bunke, Thomas Schick, Markus Spitzweck, and Andreas Thom

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2281.1 A sheaf theoretic version of Pontrjagin duality . . . . . . . . . . . . . . . . . . 2281.2 Picard stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2291.3 Duality of Picard stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2301.4 Admissible groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2321.5 T -duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

2 Sheaves of Picard categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2362.1 Picard categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2362.2 Examples of Picard categories . . . . . . . . . . . . . . . . . . . . . . . . . . 2382.3 Picard stacks and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2392.4 Additive functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2402.5 Representation of Picard stacks by complexes of sheaves of groups . . . . . . . 242

3 Sheaf theory on big sites of topological spaces . . . . . . . . . . . . . . . . . . . . . 2483.1 Topological spaces and sites . . . . . . . . . . . . . . . . . . . . . . . . . . . 2483.2 Sheaves of topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . 2483.3 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2513.4 Application to sites of topological spaces . . . . . . . . . . . . . . . . . . . . . 2613.5 Zmult-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

4 Admissibility of sheaves represented by topological abelian groups . . . . . . . . . . 2694.1 Admissible sheaves and groups . . . . . . . . . . . . . . . . . . . . . . . . . . 2694.2 A double complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2724.3 Discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2794.4 Admissibility of the groups Rn and Tn . . . . . . . . . . . . . . . . . . . . . 2864.5 Profinite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2894.6 Compact connected abelian groups . . . . . . . . . . . . . . . . . . . . . . . . 307

5 Duality of locally compact group stacks . . . . . . . . . . . . . . . . . . . . . . . . 3155.1 Pontrjagin Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3155.2 Duality and classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

6 T -duality of twisted torus bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 3236.1 Pairs and topological T -duality . . . . . . . . . . . . . . . . . . . . . . . . . . 3236.2 Torus bundles, torsors and gerbes . . . . . . . . . . . . . . . . . . . . . . . . . 3276.3 Pairs and group stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3326.4 T -duality triples and group stacks . . . . . . . . . . . . . . . . . . . . . . . . 337

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

228 U. Bunke, T. Schick, M. Spitzweck, and A. Thom

1 Introduction

1.1 A sheaf theoretic version of Pontrjagin duality

1.1.1 A character of an abelian topological group G is a continuous homomorphism

� W G ! T

from G to the circle group T . The set of all characters of G will be denoted by yG. Itis again a group under point-wise multiplication.

Assume that G is locally compact. The compact-open topology on the space ofcontinuous maps Map.G;T / induces a compactly generated topology on this space ofmaps, and hence a topology on its subset yG � Map.G;T /. The group yG equippedwith this topology is called the dual group of G.

1.1.2 An element g 2 G gives rise to a character ev.g/ 2 yyG defined by ev.g/.�/ WD�.g/ for � 2 yG. In this way we get a continuous homomorphism

ev W G ! yyG:The main assertion of Pontrjagin duality is

Theorem 1.1 (Pontrjagin duality). If G is a locally compact abelian group, then

ev W G ! yyG is an isomorphism of topological groups.

Proofs of this theorem can be found e.g. in [Fol95] or [HM98].

1.1.3 In the present paper we use the language of sheaves in order to encode thetopology of spaces and groups. Let X;B be topological spaces. The space X givesrise to a sheaf of sets X on B which associates to an open subset U � B the setX.U / D C.U;X/ of continuous maps from U to X . If G is a topological abeliangroup, then G is a sheaf of abelian groups on B .

The space X or the group G is not completely determined by the sheaf it generatesover B . As an extreme example take B D ¹�º. Then X and the underlying discretespace Xı induce isomorphic sheaves on B .

For another example, assume that B is totally disconnected. Then the sheavesgenerated by the spaces Œ0; 1� and ¹�º are isomorphic.

1.1.4 But one can do better and consider sheaves which are defined on all topolog-ical spaces or at least on a sufficiently big subcategory S � TOP. We turn S into aGrothendieck site determined by the pre-topology of open coverings (for details aboutthe choice of S see Section 3). We will see that the topology in S is sub-canonical sothat every object X 2 S represents a sheaf X 2 ShS. By the Yoneda Lemma the spaceX can be recovered from the sheaf X 2 ShS represented by X . The evaluation of thesheaf X on A 2 S is defined by X.A/ WD HomS.A;X/. Our site S will in particularcontain all locally compact spaces.

Duality for topological abelian group stacks and T -duality 229

1.1.5 We can reformulate Pontrjagin duality in sheaf theoretic terms. The circle groupT belongs to S and gives rise to a sheaf T . Given a sheaf of abelian groups F 2 ShAb Swe define its dual by

D.F / WD HomShAb S.F;T /

and observe thatD.G/ Š yG

for an abelian group G 2 S.The image of the natural pairing

F ˝Z D.F /! T

under the isomorphism

HomShAb S.F ˝Z D.F /;T / Š HomShAb S.F;HomShAb S.D.F /;T //

D HomShAb S.F;D.D.F ///

gives the evaluation mapevF W F ! D.D.F //:

Definition 1.2. We call a sheaf of abelian groups F 2 ShAb S dualizable, if the evalu-ation map evF W F ! D.D.F // is an isomorphism of sheaves.

The sheaf theoretic reformulation of Pontrjagin duality is now:

Theorem 1.3 (Sheaf theoretic version of Pontrjagin duality). IfG is a locally compactabelian group, then G is dualizable.

1.2 Picard stacks

1.2.1 A group gives rise to a category BG with one object so that the group appears asthe group of automorphisms of this object. A sheaf-theoretic analog is the notion of agerbe.

1.2.2 A set can be identified with a small category which has only identity morphisms.In a similar way a sheaf of sets can be considered as a strict presheaf of categories. Inthe present paper a presheaf of categories on S is a lax contravariant functor S! Cat.Thus a presheaf F of categories associates to each object A 2 S a category F.A/, andto each morphism f W A! B a functor f � W F.B/! F.A/. The adjective lax means,that in addition for each pair of composable morphisms f; g 2 S we have specified

an isomorphism of functors g� ı f � �f;gŠ .f ı g/� which satisfies higher associativityrelations. The sheaf of categories is called strict if these isomorphisms are identities.

1.2.3 A category is called a groupoid if all its morphisms are isomorphisms. A prestackon S is a presheaf of categories on S which takes values in groupoids. A prestack is astack if it satisfies in addition descent conditions on the level of objects and morphisms.For details about stacks we refer to [Vis05].


1.2.4 A sheaf of groups F 2 ShAb S gives rise to a prestack pBF which associates toU 2 S the groupoid pBF.U / WD BF.U /. This prestack is not a stack in general. Butit can be stackified (this is similar to sheafification) to the stack BF .

1.2.5 For an abelian group G the category BG is actually a group object in Cat. Thegroup operation is implemented by the functor BG � BG ! BG which is obvious onthe level of objects and given by the group structure of G on the level of morphisms.In this case associativity and commutativity is strictly satisfied. In a similar manner,for F 2 ShAb S the prestack pBF on S becomes a group object in prestacks on S.

1.2.6 A group G can of course also be viewed as a category G with only identitymorphisms. This category is again a strict group object in Cat. The functorG�G ! G

is given by the group operation on the level of objects, and in the obvious way on thelevel of morphisms.

1.2.7 In general, in order to define group objects in two-categories like Cat or the cat-egory of stacks on S, one would relax the strictness of associativity and commutativity.For our purpose the appropriate relaxed notion is that of a Picard category which wewill explain in detail in 2. The corresponding sheafified notion is that of a Picard stack.We let PIC S denote the two-category of Picard stacks on S.

1.2.8 A sheaf of groups F 2 SAbS gives rise to a Picard stack in two ways.First of all it determines the Picard stack F which associates to U 2 S the Picard

category F.U / in the sense of 1.2.6. The other possibility is the Picard stack BF

obtained as the stackification of the Picard prestack pBF .

1.2.9 Let A 2 PIC S be a Picard stack. The presheaf of its isomorphism classes ofobjects generates a sheaf of abelian groups which will be denoted byH 0.A/ 2 ShAb S.The Picard stack A gives furthermore rise to the sheaf of abelian groups H�1.A/ 2ShAb S of automorphisms of the unit object.

A Picard stack A 2 PIC S fits into an extension (see 2.5.12)

BH�1.A/! A! H 0.A/: (1)

1.3 Duality of Picard stacks

1.3.1 It is an essential observation by Deligne that the two-category of Picard stacksPIC S admits an interior HOMPIC S. Thus for Picard stacks A;B 2 PIC S we have aPicard stack

HOMPIC S.A;B/ 2 PICS

of additive morphisms from A to B (see 2.4).


1.3.2 Since we can consider sheaves of groups as Picard stacks in two different waysone can now ask how Pontrjagin duality is properly reflected in the language of Picardstacks. It turns out that the correct dualizing object is the stack BT 2 PIC S, and notT 2 PIC S as one might guess first. For a Picard stack A we define its dual by

D.A/ WD HOMPIC S.A;BT /:

We hope that using the same symbol for the dual sheaf and dual Picard stack does notintroduce to much confusion (see the two footnotes attached to the formulas (2) and(3) below). One can ask what the duals of the Picard stacks F and BF look like. Ingeneral we have (see 5.5)

D.BF / Š D.F /:1 (2)

One could expect thatD.F / Š BD.F /; 2 (3)

but this only holds under the condition that Ext1ShAb S.F;T / D 0 (see 5.6). This conditionis not always satisfied, e.g.

Ext1ShAb S.˚n2NZ=2Z;T / 6D 0(see 4.29). But the main effort of the present paper is made to show that

Ext1ShAb S.G;T / D 0for a large class of locally compact abelian groups (this condition is part of admissibility,see Definition 1.5).

1.3.3 Let A 2 PICS be a Picard stack. There is a natural evaluation

evA W A! D.D.A//:

In analogy to 1.2 we make the following definition.

Definition 1.4. We call a Picard stack dualizable if evA W A! D.D.A// is an equiva-lence of Picard stacks.

1.3.4 If G is a locally compact abelian group and

Ext1ShAb S.G;T / Š Ext1ShAb S.yG;T / Š 0; (4)

then the evaluation maps

evG W G ! D.D.G//; evBG W BG ! D.D.BG//

are isomorphisms by applying (2) and (3) twice, and using the sheaf-theoretic versionof Pontrjagin duality 1.3. In other words, under the condition (4) aboveG and BG aredualizable Picard stacks.

1On the right-hand sideD.F / is the dual sheaf as in 1.1.5, considered as a Picard stack as in 1.2.8.2The symbol D.F / on the left-hand side of this equation denotes the dual of the Picard stack given by

the sheaf F , whileD.F / on the right-hand side is the dual sheaf.


1.3.5 Given the structure

BH�1.A/! A! H 0.A/

of A 2 PIC S, we ask for a similar description of the dual D.A/ 2 PIC S.We will see under the crucial condition

Ext1ShAb S.H0.A/;T / Š Ext2ShAb S.H

�1.A/;T / Š 0;that D.A/ fits into

BD.H 0.A//! D.A/! D.H�1.A//:

Without the condition the description ofH�1.D.A// is more complicated, and we referto (45) for more details.

1.3.6 This discussion now leads to one of the main results of the present paper.

Definition 1.5. We call the sheaf F 2 ShAb S admissible, iff

Ext1ShAb S.F;T / Š Ext2ShAb S.F;T / Š 0:Theorem 1.6 (Pontrjagin duality for Picard stacks). If A 2 PIC S is a Picard stacksuch that H i .A/ and D.H i .A// are dualizable and admissible for i D �1; 0, then Ais dualizable.

1.4 Admissible groups

1.4.1 Pontrjagin duality for locally compact abelian groups implies that the sheaf Gassociated to a locally compact abelian group G is dualizable. In order to apply Theo-rem 1.6 to Picard stacksA 2 PIC S whose sheavesH i .A/, i D �1; 0 are represented bylocally compact abelian groups we must know for a locally compact group G whetherthe sheaf G is admissible.

Definition 1.7. We call a locally compact groupG admissible, if the sheafG is admis-sible.

1.4.2 Admissibility of a locally compact group is a complicated property. Not everylocally compact group is admissible, e.g. a discrete group of the form˚n2NZ=pZ forsome integer p is not admissible (see 4.29).

1.4.3 Admissibility is a vanishing condition

Ext1ShAb S.G;T / Š Ext2ShAb S.G;T / Š 0:This condition depends on the site S. In the present paper we shall also consider the sub-sites Slc-acyc � Slc � S of locally compact locally acyclic spaces and locally compactspaces.


For these sites the Ext-functor commutes with restriction (we verify this propertyin 3.4). Admissibility thus becomes a weaker condition on a smaller site. We will refineour notion of admissibility by saying that a group G is admissible on the site Slc (orsimilarly for Slc-acyc), if the corresponding restrictions of the extension sheaves vanish,e.g.

Ext1ShAb S.G;T /jSlc Š Ext2ShAb S.G;T /jSlc Š 0in the case Slc.

1.4.4 Some locally compact abelian groups are admissible on the site S. This appliese.g. to finitely generated groups like Z;Z=nZ, but also to Tn and Rn.

In the case of profinite groups G we need the technical assumption that it does nothave too much two-torsion and three-torsion.

Definition 1.8 (4.6). We say that the topological abelian groupG satisfies the two-threecondition, if

1. it does not admitQn2N Z=2Z as a sub-quotient,

2. the multiplication by 3 on the component G0 of the identity has finite cokernel.

We can show that a profinite abelian group which satisfies the two-three conditionis also admissible. We conjecture that it is possible to remove the condition using othertechniques.

A compact connected abelian group is divisible. Hence, if p 2 N is a prime, thenthe multiplication p W G ! G is set-theoretically surjective.

Definition 1.9. We say that G is locally topologically p-divisible, if the map p has acontinuous local section. The group G is locally topologically divisible, if it is locallytopologically p-divisable for all primes p.

For a general connected compact group which satisfies the two-three condition wecan only show that it is admissible on Slc-acyc. If it is locally topologically divisible,then it is admissible on the larger site Slc.

We think that the two-tree condition and the restriction to a locally compact site isof technical nature. The condition of local compactness enters the proof since at oneplace we want to calculate the cohomology of the sheaf Z on the space A �G using aKünneth formula. For this reason we want that A is locally compact.

As our counterexample above shows, a general (infinitely generated) discrete groupis not admissible unless we restrict to the site Slc-acyc. We do not know if the restrictionto the site Slc-acyc is really necessary for a general compact connected groups.

Using that the class of admissible groups is closed under finite products and exten-sions we get the following general theorem.

Theorem 1.10 (4.8). 1. If G is a locally compact abelian group which satisfies thetwo-three condition, then it is admissible over Slc-acyc.

2. Assume that


(a) G satisfies the two-three condition,

(b) G admits an open subgroup of the formC�Rn withC compact such thatG=C�Rn

is finitely generated

(c) the connected component of the identity of G is locally topologically divisible.

Then G is admissible over Slc.

The whole Section 4 is devoted to the proof of this theorem. This section is verylong and technical. A reader who is interested in the extension of Pontrjagin duality totopological abelian group stacks and applications to T -duality is advised to skip thissection in a first reading and to take Theorem 4.8 for granted.

1.5 T -duality

1.5.1 The aim of topological T -duality is to model the underlying topology of mirrorsymmetry in algebraic geometry and T -duality in string theory. For a detailed motiva-tion we refer to [BRS]. The objects of topological T -duality over a space B are pairs.E;G/ of a Tn-principal bundle E ! B and a gerbe G ! E with band T . A gerbewith band T over a space E is a map of stacks G ! E which is locally isomorphicto BTjE ! E. Topological T -duality associates to .E;G/ dual pairs . yE; yG/. Forprecise definitions we refer to [BRS] and 6.1.

1.5.2 The case n D 1 (n is the dimension of the fibre of E ! B) is quite easyto understand (see [BS05]). In this case every pair admits a unique T -dual (up toisomorphism, of course). The higher dimensional case is more complicated since onthe one hand not every pair admits a T -dual, and on the other hand, in general a T -dualis not unique, compare [BS05], [BRS].

1.5.3 The general idea is that the construction of a T -dual pair of .E;G/ needs thechoice of an additional structure. This structure might not exist, and in this case thereis no T -dual. On the other hand, if the additional structure exists, then it might not beunique, so that the T -dual is not unique, too.

1.5.4 The additional structure in [BRS] was the extension of the pair to a T -dualitytriple. We review this notion in 6.1.

One can also interpret the approach to T -duality via non-commutative topology[MR05], [MR06] in this way. The gerbe G ! E determines an isomorphism class ofa bundle of compact operators (by equality of the Dixmier–Douady classes). The addi-tional structure which determines a T -dual is an Rn-action on this bundle of compactoperators which lifts the Tn-action on E. Let us call a bundle of algebras of compactoperators on E together with such a Rn-action a dynamical triple.

The precise relationship between T -duality triples and dynamical triples is studiedin the thesis of Ansgar Schneider [Sch07].


1.5.5 The initial motivation of the present paper was a third choice for the additionalstructure of the pair which came to live in a discussion with T. Pantev in spring 2006.It was motivated by the analogy with some constructions in algebraic geometry, seee.g. [DP].

The starting point is the observation that a T -dual exists if and only if the restrictionof the gerbe G ! E to EjB.1/ , the restriction of E to a one-skeleton of B , is trivial.In particular, the restriction of G to the fibres of E has to be trivial. Of course, theT n-bundle E ! B is locally trivial on B , too. Therefore, locally on the base B , thesequence of maps G ! E ! B is equivalent to

.BT � Tn/jB ! TnjB ! B jB ;

where we identify spaces over B with the corresponding sheaves. The stack .BT �Tn/jB is a Picard stack.

Our proposal for the additional structure on G ! E ! B is that of a torsor overthe Picard stack .BT � Tn/jB .

1.5.6 In order to avoid the definition of a torsor over a group object in the two-categorialworld of stacks we use the following trick (see 6.2 for details). Note that a torsorX overan abelian group G can equivalently be described as an extension of abelian groups

0! G ! Up! Z! 0

so that X Š p�1.1/. We use the same trick to describe a torsor over the Picard stack.BT � Tn/jB as an extension

.BT � Tn/jB ! U ! ZjBof Picard stacks.

1.5.7 The sheaf of sections of E ! B is a torsor over Tn. Let

0! TnjB ! E ! ZjB ! 0

be the corresponding extension of sheaves of abelian groups. The filtration (1) of thePicard stack U has the form

BT jB ! U ! E;

and locally on BU Š .BT � Tn � Z/jB : (5)

1.5.8 The proposal 1.5.5 was motivated by the hope that the Pontrjagin dual D.U / ofU determines the T -dual pair. In fact, this can not be true directly since the structureof D.U / (this uses admissibility of Tn and Zn) is given locally on B by

D.U / Š .BT �BZn � Z/jB :

Here the factor Z in (5) gives rise to BT , the factor BT yields Z, and Tn yields BZn

according to the rules (2) and (3). The problematic factor is BZn in a place wherewe expect a factor Tn. The way out is to interpret the gerbe BZn as the gerbe ofRn-reductions of the trivial principal bundle Tn � B ! B .


1.5.9 We have canonical isomorphisms H 0.D.U // Š D.H�1.U // Š D.T jB/ ŠZjB and let D.U /1 � D.U / be the pre-image of ¹1ºjB� ZjB under the natural mapD.U /! H 0.D.U //. The quotient D.U /1=.BT /jB is a Zn-gerbe over B which weinterpret as the gerbe of Rn-reductions of a Tn-principal bundle yE ! B which iswell-defined up to unique isomorphism. The full structure of D.U / supplies the dualgerbe yG ! yE. The details of this construction are explained in 6.4.

1.5.10 In this way we use Pontrjagin duality of Picard stacks in order to construct aT -dual pair to .E;G/. Schematically the picture is

.E;G/1Ý U

2Ý D.U /3Ý . yE; yG/;

where the steps are as follows:

1. choice of the structure on G of a torsor over .BT � Tn/jB

2. Pontrjagin duality D.U / WD HomPIC S.U;BT /

3. Extraction of the dual pair from D.U / as explained in 1.5.9.

1.5.11 Consider a pair .E;G/. In Subsection 6.4 we provide two constructions:

1. The construction ˆ starts with the choice of a T -duality triple extending .E;G/and constructs the structure of a torsor over .BT � Tn/jB on G.

2. The construction ‰ starts with the structure on G of a torsor over .BT �Tn/jBand constructs an extension of .E;G/ to a T -duality triple.

Our main result Theorem 6.23 asserts that the constructions ‰ and ˆ are inverses toeach other. In other words, the theories of topological T -duality via Pontrjagin dualityof Picard stacks and via T -duality triples are equivalent.

2 Sheaves of Picard categories

2.1 Picard categories

2.1.1 In a cartesian closed category one can define the notion of a group object in thestandard way. It is given by an object, a multiplication, an identity, and an inversionmorphism. The group axioms can be written as a collection of commutative diagramsinvolving these morphisms.

Stacks on a site S form a two-cartesian closed two-category. A group object in atwo-cartesian closed two-category is again given by an object and the multiplication,identity and inversion morphisms. In addition each of the commutative diagrams fromthe category case is now filled by a two-morphisms. These two-morphisms must satisfyhigher associativity relations.

Instead of writing out all these relations we will follow the exposition of DeligneSGA4, exposé XVIII [Del], which gives a rather effective way of working with group


objects in two-categories. We will not be interested in the most general case. For ourpurpose it suffices to consider a notion which includes sheaves of abelian groups andgerbes with band given by a sheaf of abelian groups. We choose to work with sheavesof strictly commutative Picard categories.

2.1.2 Let C be a category,

F W C � C ! C

be a bi-functor, and

� W F.F.X; Y /;Z/ �! F.X; F.Y;Z//

be a natural isomorphism of tri-functors.

Definition 2.1. The pair .F; �/ is called an associative functor if the following holds.For every family .Xi /i2I of objects of C let e W I ! M.I/ denote the canonicalmap into the free monoid (without identity) on I . We require the existence of a

map F W M.I/ ! Ob.C / and isomorphisms ai W F .e.i// �! Xi , and isomorphisms

ag;h W F .gh/ �! F.F .g/; F .h// such that the following diagram commutes:

F .f .gh//af;gh

�� F.F .f /; F .gh//ag;h

�� F.F .f /; F.F .g/; F .h//

F ..fg/h/afg;h

�� F.F .fg/; F .h//af;g

�� F.F.F .f /; F .g//; F .h//.

�

��

(6)

2.1.3 Let .F; �/ be as above. Let in addition be given a natural transformation ofbi-functors

� W F.X; Y /! F.Y;X/:

Definition 2.2. .F; �; �/ is called a commutative and associative functor if the followingholds. For every family .Xi /i2I of objects in C let e W I ! N.I / denote the canonicalmap into the free abelian monoid (without identity) on I . We require that there exist

F W N.I /! C , isomorphisms ai W F .e.i//! Xi and isomorphisms ag;h W F .gh/ �!F.F .g/; F .h// such that (6) and

F .gh/ag;h

�� F.F .g/; F .h//

�

��

F .hg/ah;g

�� F.F .h/; F .g//

(7)

commute.


2.1.4 We can now define the notion of a strict Picard category. Instead of F will usethe symbol “C”.

Definition 2.3. A Picard category is a groupoid P together with a bi-functor CW P �P ! P and natural isomorphisms � and � as above such that .C; �; �/ is a commutativeand associative functor, and such that for all X 2 P the functor Y 7! X C Y is anequivalence of categories.

2.2 Examples of Picard categories

2.2.1 Let G be an abelian group. We consider G as a category with only identitymorphisms. The addition CW G �G ! G is a bi-functor. For � and � we choose theidentity transformations. Then .G;C; �; �/ is a strict Picard category which we willdenote again by G.

2.2.2 Let us now consider the abelian group G as a category BG with one object �and MorBG.�;�/ WD G. We let CW BG � BG ! BG be the bi-functor which acts asaddition MorBG.�/�MorBG.�/! MorBG.�/. For � and � we again choose the identitytransformations. We will denote this strict Picard category .BG;C; �; �/ shortly by BG.

2.2.3 LetG;H be abelian groups. We define the category EXT.G;H/ as the categoryof short exact sequences

E W 0! H ! E ! G ! 0:

Morphisms in EXT.G;H/ are isomorphisms of complexes which reduce to the identityon G and H . We define the bi-functor CW EXT.G;H/ � EXT.G;H/! EXT.G;H/as the Baer addition

E1 C E2 W 0! H ! F ! G ! 0;

where F WD zF=H , zF � E1 � E2 is the pre-image of the diagonal in G � G, andthe action of H on zF is induced by the anti-diagonal action on E1 � E2. The asso-ciativity morphism � W .E1 C E2/C E3 ! E1 C .E2 C E3/ is induced by the canon-ical isomorphism .E1 � E2/ � E3 ! E1 � .E2 � E3/. Finally, the transformation� W E1CE2 ! E2CE1 is induced by the flipE1�E2 ! E2�E1. The triple .C; �; �/defines on EXT.G;H/ the structure of a Picard category.

2.2.4 Let G be a topological abelian group, e.g. G WD T . On a space B we considerthe category ofG-principal bundles BG.B/. Given twoG-principal bundlesQ! B ,P ! B we can define a new G-principal bundle Q ˝G P WD Q �B P=G. Thequotient is taken by the anti-diagonal action, i.e. we identify .q; p/ � .qg; pg�1/. TheG-principal structure on Q˝G P is induced by the action Œq; p�g WD Œq; pg�.

We define the associativity morphism

� W .Q˝G P /˝G R! Q˝G .P ˝G R/


as the map ŒŒq; p�; r� ! Œq; Œp; r��. Finally we let � W Q ˝G P ! P ˝G Q be givenby Œq; p�! Œp; q�.

We claim that .˝G ; �; �/ is a strict Picard category. Let I be a collection of objectsof BG.B/. We choose an ordering of I . Then we can write each element of j 2 N.I /in a unique way as ordered product f D P1P2 : : : Pr with P1 � P2 � Pr . We

defineF .f / WD P1˝G .P2˝G . ˝GPr/ /. We let aP W F .P / �! P ,P 2 I , be

the identity. Furthermore, for f; g 2 N.I / we define af;g W F .fg/ �! F .f /˝B F .g/as the map induced the permutation which puts the factors of f and g in order (and sothat the order to repeated factors is not changed). Commutativity of (6) is now clear.In order to check the commutativity of (7) observe that � W Q˝GQ! Q˝GQ is theidentity, as Œqg; q� 7! Œq; qg� D Œqg; q�.

2.3 Picard stacks and examples

2.3.1 We consider a site S. A prestack on S is a lax associative functor P W Sop !groupoids, where from now on by a groupoid we understand an essentially small cate-gory3 in which all morphisms are isomorphisms. Lax associative means that as a part ofthe data for each composable pair of maps f; g in S there is an isomorphism of functors

lf;g W P.f / ı P.g/ �! P.g ı f /, and these isomorphisms satisfy higher associativityrelation. A prestack is a stack if it satisfies the usual descent conditions one the levelof objects and morphisms. For a reference of the language of stacks see e.g. [Vis05].

Definition 2.4. A Picard stack P on S is a stack P together with an operationCW P � P ! P and transformations � , � which induce for each U 2 S the structureof a Picard category on P.U /.

2.3.2 Let ShAbS denote the category of sheaves of abelian groups on S. Extending theexample 2.2.1 to sheaves we can view each object F 2 ShAb S as a Picard stack, whichwe will again denote by F .

2.3.3 We can also extend Example 2.2.2 to sheaves. In this way every sheaf F 2 ShAbSgives rise to a Picard stack BF .

2.3.4 We now extend the example 2.2.3 to sheaves. First of all note that one can definea Picard category EXT.G ;H / of extensions of sheaves 0 ! H ! E ! G ! 0 as adirect generalization of 2.2.3. Then we define a prestack EXT.G ;H / which associatesto U 2 S the Picard category EXT.G ;H /.U / WD EXT.GjU ;HjU /. One checks thatEXT.G ;H / is a stack.

2.3.5 Let G 2 ShAbS be a sheaf of abelian groups. We consider G as a group object inthe category ShS of sheaves of sets on S. A G -torsor is an object T 2 ShS with an actionof G such that T is locally isomorphic to G . We consider the category of G -torsorsTors.G / and their isomorphisms. We define the functor CW Tors.G / � Tors.G / !

3Later, e.g. in 2.5.3, we need that the isomorphism classes in a groupoid form a set.


Tors.G / by T1 C T2 WD T1 � T2=G , where we take the quotient by the anti-diagonalaction. The structure of a G -torsor is induced by the action of G on the second factor T2.We let � and � be induced by the associativity transformation of the cartesian productand the flip, respectively. With these structures Tors.G / becomes a Picard category.

We define a Picard stack T ors.G / by localization, i.e. we set T ors.G /.U / WDTors.GjU /.

We have a canonical map of Picard stacks

U W EXT.Z;G /! T ors.G /

which maps the extension E W 0! G ! E�! Z! 0 to theG-torsorU.E/ D ��1.1/.

Here Z denotes the constant sheaf on S with value Z. One can check that U is anequivalence of Picard stacks (see 2.4.3 for precise definitions).

2.3.6 The example 2.2.4 has a sheaf theoretic interpretation. We assume that S is somesmall subcategory of the category of topological spaces which is closed under takingopen subspaces, and such that the topology is induced by the coverings of spaces byfamilies of open subsets.

Let G be a topological abelian group. We have a Picard prestack pBG on Swhich associates to each space B 2 S the Picard category pBG.B/ of G-principalbundles on B . As in 2.2.4 the monoidal structure on pBG.B/ is given by the tensorproduct ofG-principal bundles (this uses thatG is abelian). We now define BG as thestackification of pBG.

The topological groupG gives rise to a sheafG 2 ShAb S which associates toU 2 Sthe abelian group G.U / WD C.U;G/. We have a canonical transformation

� W BG ! T ors.G/:

It is induced by a transformation p� W pBG ! T ors.G/. We describe the functorp�B W pBG.B/ ! T ors.G/.B/ for all B 2 S. Let E ! B be a G-principal bun-dle. Then we define p�B.E/ 2 Tors.GjB/ to be the sheaf which associates to each.� W U ! B/ 2 S=B the set of sections of ��E ! U .

One can check that � is an equivalence of Picard stacks (see 2.4.3 for precisedefinitions).

2.4 Additive functors

2.4.1 We shall now discuss the notion of an additive functor between Picard stacksF W P1 ! P2.

Definition 2.5. An additive functor between Picard categories is a functorF W P1 ! P2

and a natural transformationF.xCy/ �! F.x/CF.y/ such that the following diagramscommute:


1.

F.x C y/ ��

F.�/

��

F.x/C F.y/�

��

F.y C x/ �� F.y/C F.x/;

2.

F..x C y/C z/ ��

F.�/

��

F.x C y/C F.z/ �� .F.x/C F.y//C F.z/�

��

F.x C .y C z// �� F.x/C F.y C z// �� F.x/C .F.y/C F.z//.

Definition 2.6. An isomorphism between additive functors u W F ! G is an isomor-phism of functors such that

F.x C y/ uxCy��

��

G.x C y/

��

F.x/C F.y/ uxCuy�� G.x/CG.y/

commutes.

Definition 2.7. We let Hom.P1; P2/ denote the groupoid of additive functors from P1to P2. By PIC we denote the two-category of Picard categories.

2.4.2 The groupoid Hom.P1; P2/ has a natural structure of a Picard category. We set

.F1 C F2/.x/ WD F1.x/C F2.x/

and define the transformation

.F1 C F2/.x C y/! .F1 C F2/.x/C .F1 C F2/.y/such that

.F1 CF2/.xC y/ �� .F1 CF2/.x/C .F1 CF2/.y/

F1.xC y/CF2.xC y/ �� .F1.x/CF1.y//C .F2.x/CF2.y// ˛ı�ı˛ �� F1.x/CF2.x/CF1.y/CF2.y/

commutes. The associativity and commutativity constraints are induced by those ofP2.


2.4.3 Let now P1; P2 be two Picard stacks on S.

Definition 2.8. An additive functor F W P1 ! P2 is a morphism of stacks togetherwith a two-morphism

P1 � P1 C��

F�F��

��P1

F

��

P2 � P2 C�� P2

which induces for eachU 2 S an additive functorFU W P1.U /! P2.U / of Picard cate-gories. An isomorphism between additive functors u W F1 ! F2 is a two-isomorphismof morphisms of stacks which induces for each U 2 S an isomorphism of additivefunctors uU W F1;U ! F2;U .

As in the case of Picard categories, the additive functors between Picard stacks formagain a Picard category.

Definition 2.9. We let Hom.P1; P2/ denote the groupoid of additive functors from P1,P2.We get a two-category PIC.S/ of Picard stacks on S.

Definition 2.10. We let HOM.P1; P2/ denote the Picard category of additive functorsbetween Picard stacks P1 and P2.

2.4.4 Let P;Q be Picard stacks on the site S. By localization we define a Picard stackHOM.P;Q/.

Definition 2.11. The Picard stack HOM.P;Q/ is the sheaf of Picard categories givenby

HOMPIC.S/.P;Q/.U / WD HOM.PjU ;QjU /:A priori this describes a prestack, but one easily checks the stack conditions.

2.5 Representation of Picard stacks by complexes of sheaves of groups

2.5.1 There is an obvious notion of a Picard prestack. Furthermore, there is an asso-ciated Picard stack construction a such that for a Picard prestack P and a Picard stackQ we have a natural equivalence

Hom.aP;Q/ Š Hom.P;Q/ (8)

2.5.2 Let S be a site. We follow Deligne, SGA 4.3, Expose XVIII, [Del]. Let C.S/ bethe two-category of complexes of sheaves of abelian groups

K W 0! K�1 d! K0 ! 0

which live in degrees �1; 0. Morphisms in C.S/ are morphisms of complexes, andtwo-isomorphisms are homotopies between morphisms.

To such a complex we associate a Picard stack ch.K/ on S as follows. We firstdefine a Picard prestack pch.K/.


1. For U 2 S we define the set of objects of pch.K/.U / as K0.U /.

2. For x; y 2 K0.U / we define the set of morphisms by

Hompch.K/.U /.x; y/ WD ¹f 2 K�1.U / j df D x � yº:

3. The composition of morphisms is addition in K�1.U /.

4. The functorCW pch.K/.U /� pch.K/.U /! pch.K/.U / is given by addition ofobjects and morphisms. The associativity and the commutativity constraints arethe identities.

Definition 2.12. We define ch.K/ as the Picard stack associated to the prestack pch.K/,i.e. ch.K/ WD apch.K/.

2.5.3 If P is a Picard stack, then we can define a presheaf pH 0.P / which associatesto U 2 S the group of isomorphism classes of P.U /. We let H 0.P / be the associatedsheaf. Furthermore we have a sheaf H�1.P / which associates to U 2 S the group ofautomorphisms Aut.eU /, where eU 2 P.U / is a choice of a neutral object with respectto C. Since the neutral object eU is determined up to unique isomorphism, the groupAut.eU / is also determined up to unique isomorphism. Note that Aut.eU / is abelian, asshows the following diagram

e C eŠ��

idCf�� e C e

Š��

gCid�� e C e

Š��

ef

�� eg

�� e

and the fact that .idC f / ı .g C id/ D g C f D .g C id/ ı .idC f /.IfV ! U is a morphism in S, then we have a unique isomorphismf W .eU /jV �! eV

which induces a group homomorphism f ı ı f �1 W Aut.eU /jV ! Aut.eV /. Thishomomorphism is the structure map of the sheaf H�1.P / which is determined up tounique isomorphism in this way.

We now observe that

H�1.ch.K// Š H�1.K/; H 0.ch.K// Š H 0.K/: (9)

2.5.4 A morphism of complexes f W K ! L in C.S/, i.e. a diagram

0 �� K�1

f �1

��

dK �� K0

f 0

��

�� 0

0 �� L�1dL �� L0 �� 0,

induces an additive functor of Picard prestacks pch.f / W pch.K/! pch.L/ and, by thefunctoriality of the associated stack construction, an additive functor

ch.f / W ch.K/! ch.L/:

In view of (9), it is an equivalence if and only if f is a quasi isomorphism.


2.5.5 We consider two morphisms f; g W K ! L of complexes in C.S/. A homotopyH W f ! g is a morphism of sheaves H W K0 ! L�1 such that f 0 � g0 D dL ıHand f �1�g�1 D H ıdK . It is easy to see thatH induces an isomorphism of additivefunctors pch.H/ W pch.f /! pch.g/ and therefore

ch.H/ W ch.f /! ch.g/

by the associated stack construction and (8).One can show that the isomorphisms pch.f / ! pch.g/ correspond precisely to

homotopies H W f ! g. This implies that the morphisms ch.f / ! ch.g/ alsocorrespond bijectively to these homotopies.

2.5.6 We have the following result.

Lemma 2.13 ([Del, 1.4.13]). 1. For every Picard stack P there exists a complexK 2 C.S/ such that P Š ch.K/.

2. For every additive functor F W ch.K/! ch.L/ there exists a quasi isomorphismk W K 0 ! K and a morphism l W K 0 ! L in C.S/ such that F Š ch.l/ ı ch.k/�1.

2.5.7 Let PIC[.S/ be the category of Picard stacks obtained from the two-categoryPIC S by identifying isomorphic additive functors.

Proposition 2.14 ([Del, 1.4.15]). The construction ch gives an equivalence of cate-gories

DŒ�1;0�.ShAb S/! PIC[.S/;

where DŒ�1;0�.ShAb S/ is the full subcategory of DC.ShAb S/ of objects whose coho-mology is trivial in degrees 62 ¹�1; 0º.Lemma 2.15 ([Del, 1.4.16]). Let K;L 2 C.S/ and assume that L�1 is injective.

1. pch.L/ is already a stack.

2. For every morphism F 2 HomPIC.S/.ch.K/; ch.L// there exists a morphismf 2 HomC.ShAb S/.K;L/ such that ch.f / Š F .

Let C 0.S/ � C.S/ denote the full sub-two-category of complexes 0 ! K�1 !K0 ! 0 with K�1 injective.

Lemma 2.16 ([Del, 1.4.17]). The construction ch induces an equivalence of the two-categories PIC.S/ and C 0.S/.

2.5.8 Finally we give a characterization of the Picard stack Hom.ch.K/; ch.L//.

Lemma 2.17 ([Del, 1.4.18.1]). Assume that L�1 is injective. Then we have an equiv-alence

ch.��0HomShAb S.K;L//�! HOMPIC.S/.ch.K/; ch.L//:


Lemma 2.18. Let K;L 2 C.S/. Then we have an isomorphism

H i .HOMPIC.S/.ch.K/; ch.L/// Š RiHomShAb S.K;L/

for i D �1; 0.

Proof. First observe that by the discussion above the left hand side, and by the definitionofRHom the right side both only depend on the quasi-isomorphism type of the complexL of length 2. Without loss of generality we can therefore assume thatL�1 is injective.

We now choose an injective resolution I W 0! L�1 ! I 0 ! I 1 : : : of L startingwith the choice of an embeddingL0 ! I 0. Then we have Hom.K; I / Š RHom.K;L/.We now observe that H iHom.K; I / Š H iHom.K;L/ for i D 0;�1. While the casei D �1 is obvious, for i D 0 observe that a 0-cycle in Hom.K; I / is a morphism ofcomplexes and necessarily factors over L! I . We thus have for i 2 ¹�1; 0º

RiHom.K;L/ Š H iHom.K;L/ Š H i .Hom.ch.K/; ch.L///:

2.5.9 For A;B 2 ShAb S we have canonical isomorphisms

Ext2ShAb S.B;A/ Š R0HomShAb S.B;AŒ2�/ Š HomD.ShAb S/.B;AŒ2�/:

In the following we recall two eventually equivalent ways how an exact complex

K W 0! A! X ! Y ! B ! 0

represents an element

Y.K/ 2 HomD.ShAb S/.B;AŒ2�/ Š Ext2ShAb S.B;A/

(the letter Y stands for Yoneda who investigated this construction first). Let KA be thecomplex

KA W 0! X ! Y ! B ! 0;

where B sits in degree 0. The obvious inclusion ˛ W AŒ2� ! KA induced by A ! X

is a quasi-isomorphism. Furthermore, we have a canonical map ˇ W B ! KA. Theelement Y.K/ 2 HomD.ShAb S/.B;AŒ2�/ is by definition the composition

Y.K/ W B ˇ!KA˛�1

��! AŒ2�: (10)

We can also consider the complex KB given by

0! A! X ! Y ! 0

where A is in degree �2. The projection Y ! B induces a quasi-isomorphism W KB ! B . We furthermore have a canonical map ı W KB ! AŒ2�. We consider thecomposition Y 0.K/ 2 HomD.ShAb S/.B;AŒ2�/

Y 0.K/ W B ��1

��!KBı! AŒ2�: (11)


Lemma 2.19. In HomD.ShAb S/.B;AŒ2�/ we have the equality

Y.K/ D Y 0.K/:

Proof. We consider the morphism of complexes � W KB !KA given by obvious mapsin the diagram

X �� Y �� B

A

��

�� X ��

��

Y .

��

It fits into

Bˇ

�� KA AŒ2�˛

��

B KB

�

��

�� ı �� AŒ2�.

It suffices to show that the two squares commute. In fact we will show that � ishomotopic to ˇ ı and ˛ ı ı. Note that � � ˇ ı is given by

X �� Y �� B

A

��

�� X

id

��

��

��

Y .

0

��

0

��

The dotted arrows in this diagram gives the zero homotopy of this difference.Similarly � � ˛ ı ı is given by

X �� Y �� B

A

0

��

�� X

0

��

��

��

Y ,

��

id

��

and we have again indicated the required zero homotopy.

2.5.10 The equivalence between the two-categories PIC.S/ andC 0.S/ (see 2.16) allowsus to classify equivalence classes of Picard stacks with fixed H i .P / Š Ai , i D �1; 0,Ai 2 ShAb S. An equivalence of such Picard stacks is an equivalence which inducesthe identity on the cohomology.

Lemma 2.20. The set ExtPIC.S/.A0; A�1/ of equivalence classes of Picard stacks Pwith given isomorphisms H i .P / Š Ai , i D 0;�1, Ai 2 ShAb S is in bijection withExt2ShAb S.A0; A�1/.

Proof. We define a map

� W ExtPIC.S/.A0; A�1/! Ext2ShAb S.A0; A�1/ (12)


as follows.Consider an exact complex

K W 0! A�1 ! K�1 ! K0 ! A0 ! 0

such that K�1 is injective, and let K 2 C.S/ be the complex

0! K�1 ! K0 ! 0

Then we define �.ch.K// WD Y.K/.We must show that � is well-defined. Indeed, if ch.K/ Š ch.L/, then (see 2.16)

K Š L by an equivalence which induces the identity on the level of cohomology. Itfollows that Y.K/ D Y.L/.

Since ch W C 0.S/ ! PIC[.S/ is an equivalence, hence in particular surjectiveon the level of equivalence classes, the map � is well-defined. Since every ele-ment of Ext2ShAb S.A�1; A0/ can be written as Y.K/ for a suitable complex K asabove we conclude that � is surjective. If �.ch.K// Š �.ch.L//, then there existsM 2 C.S/ with given isomorphisms H i .M/ Š Ai together with quasi-isomorphismsK M ! L inducing the identity on cohomology. But this diagram induces anequivalence ch.K/ Š ch.M/ Š ch.L/.

2.5.11 We continue with a further description of BF for F 2 ShAb S, compare 1.2.8.Note that the sheaf of groups of automorphisms of BF is F , whereas the group ofobjects is trivial. Therefore we can represent BF as ch.F Œ1�/, where F Œ1� is thecomplex 0! F ! 0! 0 concentrated in degree �1.

2.5.12 Let P 2 PIC.S/ be a Picard stack and G � H�1.P /. Let us assume thatP D ch.K/ for a suitable complex K W 0 ! K�1 ! K0 ! 0. We have a naturalinjection G ,! ker.K�1 ! K0/. We consider the quotient xK�1 defined by the exactsequence 0 ! G ! K�1 ! xK�1 ! 0 and obtain a new Picard stack xP Š ch. xK/,where xK W 0 ! xK�1 ! K0 ! 0. The following diagram represents a sequence ofmorphisms of Picard stacks

BG ! P ! xP ;

0 �� G ��

��

0

��

�� 0

0 �� K�1 ��

��

K0

��

�� 0

0 �� xK�1 �� K0 �� 0.

The Picard stack xP can be considered as a quotient of P by BG. We will employ thisconstruction in 6.4.5.


3 Sheaf theory on big sites of topological spaces

3.1 Topological spaces and sites

3.1.1 In this paper a topological space will always be compactly generated and Haus-dorff. We will define categorical limits and colimits in the category of compactlygenerated Hausdorff spaces. Furthermore, we will equip mapping spaces Map.X; Y /with the compactly generated topology obtained from the compact-open topology. Inthis category we have the exponential law

Map.X � Y;Z/ Š Map.X;Map.Y;Z//:

By Map.X; Y /ı we will denote the underlying set. For details on this convenientcategory of topological spaces we refer to [Ste67].

3.1.2 The sheaf theory of the present paper refers to the Grothendieck site S. Theunderlying category of S is the category of compactly generated topological Hausdorffspaces. The covering families of a space X 2 S are coverings by families of opensubsets.

We will also need the sites Slc and Slc-acyc given by the full subcategories of locallycompact and locally compact locally acyclic spaces (see 3.4.6).

3.1.3 We let Pr S and ShS denote the category of presheaves and sheaves of sets on S.Then we have an adjoint pair of functors

i] W Pr S, ShS W iwhere i is the inclusion of sheaves into presheaves, and i] is the sheafification functor.If F 2 Pr S, then sometimes we will write F ] WD i]F .

As before, by PrAb S and ShAb S we denote the categories of presheaves and sheavesof abelian groups.

3.2 Sheaves of topological groups

3.2.1 In this subsection we collect some general facts about sheaves generated byspaces and topological groups. We formulate the results for the site S. But they remaintrue if one replaces S by Slc or Slc-acyc.

3.2.2 Every object X 2 S represents a presheaf X 2 Pr S of sets defined by

S 3 U 7! X.U / WD HomS.U;X/ 2 Sets

on objects, and by

HomS.U; V / 3 f 7! f � W X.V /! X.U /

on morphisms.


Lemma 3.1. X is a sheaf.

Proof. Straightforward.

Note that a topology is called sub-canonical if all representable presheaves aresheaves. Hence S carries a sub-canonical topology.

3.2.3 We will also need the relative version. For Y 2 S we consider the relative siteS=Y of spaces over Y . Its objects are morphisms A ! Y , and its morphisms arecommutative diagrams

A

��

��

�� B

��

��

Y .

The covering families of A! Y are induced from the coverings of A by open subsets.An object X ! Y 2 S=Y represents the presheaf X ! Y 2 Pr S=Y (by a similar

definition as in the absolute case 3.2.2). The induced topology on S=Y is again sub-canonical. In fact, we have the following lemma.

Lemma 3.2. For all .X ! Y / 2 S the presheaf X ! Y is a sheaf.

Proof. Straightforward.

3.2.4 Let I be a small category and X 2 SI . Then we have

lim i2IX.i/ Š lim i2IX.i/:

One can not expect a similar property for arbitrary colimits. But we have the followingresult.

Lemma 3.3. Let I be a directed partially ordered set and X 2 SI be a direct systemof discrete sets such that X.i/ ! X.j / is injective for all i � j . Then the canonicalmap

colim i2IX.i/! colim i2IX.i/

is an isomorphism.

Proof. Let pcolim i2IX.i/ denote the colimit in the sense of presheaves. Then we havean inclusion

pcolim i2IX.i/ ,! colim i2IX.i/

(this uses injectivity of the structure maps of the system X ). Since the target is a sheafand sheafification preserves injections it induces an inclusion

colim i2IX.i/ ,! colim i2IX.i/: (13)


Let nowA 2 S and f 2 colim i2IX.i/.A/DHomS.A; colim i2IX.i//. As colim i2IX.i/is discrete andf is continuous the family of subsets ¹f �1.x/ � A j x 2 colim i2IX.i/ºis an open covering of A by disjoint open subsets. The family

Yx

fjf �1.x/ 2Yx

pcolim i2IX.i/.f �1.x//

represents a section overA of the sheafification colim i2IX.i/ of pcolim i2IX.i/whichof course maps to f under the inclusion (13). Therefore (13) is also surjective.

3.2.5 If G 2 S is a topological abelian group, then HomS.U;G/ WD G.U / has thestructure of a group by point-wise multiplications. In this case G is a sheaf of abeliangroups G 2 ShAb S.

3.2.6 We can pass back and forth between (pre)sheaves of abelian groups and(pre)sheaves of sets using the following adjoint pairs of functors

pZ.: : : / W Pr S, PrAbS W F ; Z.: : : / W ShS, ShAbS W F :The functor F forgets the abelian group structure. The functors pZ.: : : / and Z.: : : /are called the linearization functors. The presheaf linearization associates to a presheafof sets H 2 Pr S the presheaf pZ.H/ 2 PrAb S which sends U 2 S to the free abeliangroup Z.H/.U / generated by the setH.U /. The linearization functor for sheaves is thecomposition Z.: : : / WD i]ı pZ.: : : / of the presheaf linearization and the sheafification.

Lemma 3.4. Consider an exact sequence of topological groups in S

1! G ! H ! L! 1:

Then 1! G ! H ! L is an exact sequence of sheaves of groups.The map of topological spaces H ! L has local sections if and only if

1! G ! H ! L! 1

is an exact sequence of sheaves of groups.

Proof. Exactness at G and H is clear.Assume the existence of local sections of H ! L. This implies exactness at L.On the other hand, evaluating the sequence at the object L 2 S we see that the

exactness of the sequence of sheaves implies the existence of local sections toH ! L.

3.2.7 For sheaves G;H 2 ShS we define the sheaf

HomSh S.G;H/ 2 ShS; HomSh S.G;H/.U / WD HomSh S.GjU ;HjU /:

Again, this a priori defines a presheaf, but one checks the sheaf conditions in a straight-forward manner.


3.2.8 If X;H 2 S and H is in addition a topological abelian group, then Map.X;H/is again a topological abelian group. For a topological abelian group G 2 S we letHomtop-Ab.G;H/ � Map.G;H/ be the closed subgroup of homomorphisms. Recallthe construction of the internal HomShAb S.: : : ; : : : /, compare 3.2.7.

Lemma 3.5. Assume that G;H 2 S are topological abelian groups. Then we have acanonical isomorphism of sheaves of abelian groups

e W Homtop-Ab.G;H/�! HomShAb S.G;H/:

Proof. We first describe the morphism e. Let � 2 Hom.G;H/.U /. We define theelement e.�/ 2 HomShAb S.G;H/.U /, i.e. a morphism of sheaves e.�/ W GjU ! H jU ,such that it sends f 2 G.� W V ! U/ to ¹V 3 v 7! �.�.v//f .v/º2 H.� W V ! U/.

Let us now describe the inverse e�1. Let 2 HomShAb S.G;H/.U / be given. Since.prU W U �G ! U/ 2 S=U it gives rise to a map

G W G.prU W U �G ! U/! H.prU W U �G ! U/:

We now consider .prG W U � G ! G/ 2 G.prU W U � G ! U/ and set �G WD G.prG/ 2 H.prU W U � G ! U/ D HomS.U � G;H/. We now invoke the ex-

ponential law isomorphism exp W HomS.U �G;H/ �! HomS.U;Map.G;H//. Since was a homomorphism and using the sheaf property, we actually get an elemente�1. / WD exp.�G/ 2 HomS.U;Homtop-Ab.G;H// D Homtop-Ab.G;H/.U /. (De-tails of the argument for this fact are left to the reader.)

If S is replaced by one of the sub-sites Slc or Slc-acyc, then it may happen thatHomtop-Ab.G;H/ does not belong to the sub-site. In this case Lemma 3.5 remains trueif one interprets Homtop-Ab.G;H/ as the restriction of the sheaf on S represented byHomtop-Ab.G;H/ to the corresponding sub-site.

3.3 Restriction

3.3.1 In this subsection we prove a general result in sheaf theory (Proposition 3.12)which is probably well-known, but which we could not locate in the literature. Let S bea site. For Y 2 S we can consider the relative site S=Y . Its objects are maps .U ! Y /

in S, and its morphisms are diagrams

U

��

��

�� V

��

��

Y .

The covering families for S=Y are induced by the covering families of S, i.e. for.U ! Y / 2 S=Y a covering family � WD .Ui ! U/i2I induces in S=Y a covering


family

�jY WD

0BB@Ui

��

��

�� U

��

��

Y

1CCAi2I

:

3.3.2 There is a canonical functorf W S=Y ! S given on objects byf .U ! Y / WD U .It induces adjoint pairs of functors

f� W ShS=Y , ShS W f �; pf� W Pr S=Y , Pr S W pf �:We will often write f �.F / DW FjY for the restriction functor. For F 2 Pr S it is givenin explicit terms by

FjY .U ! Y / WD F.U /:The functor f � is in fact the restriction of pf � to the subcategory of sheaves ShS �Pr S. If F 2 ShS, then one must check that pf �F is a sheaf. To this end note that thedescent conditions for pf �F with respect to the induced coverings described in 3.3.1immediately follow from the descent conditions for F .

3.3.3 Let us now study how the restriction acts on representable sheaves. We assumethat S has finite products. Let X; Y 2 S.

Lemma 3.6. If S has finite products, then we have a natural isomorphism

X jY Š X � Y ! Y :

Proof. By the universal property of the product for � W U ! Y we have

X � Y ! Y .U ! Y / D ¹h 2 HomS.U;X � Y / j prY ı h D �ºŠ HomS.U;X/

D X.U /D X jY .U ! Y /:

3.3.4 Let i] W Pr S! ShS and i]Y W Pr S=Y ! ShS=Y be the sheafification functors.

Lemma 3.7. We have a canonical isomorphism f � ı i] Š i]Y ı pf �.Proof. The argument is similar to that in the proof of [BSS, Lemma 2.31]. The mainpoint is that the sheafifications i]F.U / and i]YFjY .U ! Y / can be expressed interms of the category of covering families covS.U / and covS=Y .U ! Y /. Comparedwith [BSS, Lemma 2.31] the argument is simplified by the fact that the induction ofcovering families described in 3.3.1 induces an isomorphism of categories covS.U /!covS=Y .U ! Y /.


3.3.5 Recall that a functor is called exact if it commutes with limits and colimits.

Lemma 3.8. The restriction functor f � W ShS! ShS=Y is exact.

Proof. The functor f � is a right-adjoint and therefore commutes with limits. Sincecolimits of presheaves are defined object-wise, i.e. for a diagram

C ! Pr S; c 7! Fc

of presheaves we have

. pcolimc2CFc/.U / D colimc2CFc.U /;

it follows from the explicit description of pf �, that it commutes with colimits ofpresheaves. Indeed, for a diagram of sheaves F W C ! ShS we have

colimc2CFc D i] pcolimc2CFc :

By Lemma 3.7 we get f �colimc2CFc Š f �i] pcolimc2CFc Š i]Y pf � pcolimc2CFc Ši] pcolimc2C

pf �Fc Š colimc2Cf�Fc :

3.3.6 Let H 2 ShS and X 2 S.

Lemma 3.9. If S has finite products, then for U 2 S we have a natural bijection

HomSh S.X;H/.U / Š H.U �X/:Proof. We have the following chain of isomorphisms

HomSh S.X;H/.U / Š HomSh S=U .X jU ;HjU /Š HomSh S=U .U �X ! U ;HjU / (by Lemma 3.6)

Š H.U �X/:We will need the explicit description of this bijection. We define a map

‰ W HomSh S.X;H/.U /! H.U �X/as follows. An element h 2 HomSh S=U .X;H/.U / D HomSh S.X jU ;HjU / induces a

map Qh W X jU .U � X ! U/ ! HjU .U � X ! U/. Now we have X jU .U � X !U/ D X.U � X/ D Map.U � X;X/ and HjU .U � X ! U/ D H.U � X/. LetprX W U �X ! X be the projection. We define ‰.h/ WD Qh.prX /.

We now defineˆ W H.U�X/! HomSh S.X;H/.U /. Letg 2 H.U�X/. For each.e W V ! U/ 2 S=U we must define a map Og.V!U/ W X jU .V ! U/! HjU .V ! U/.Note that X jU .V ! U/ D X.V / D Map.V;X/ı . Let x 2 Map.V;X/. Then we havean induced map .e; x/ W V ! U � X . We define ‰.g/ WD H.e; x/.g/. One checksthat this construction is functorial in e and therefore defines a morphism of sheaves.

A straight forward calculation shows that ‰ and ˆ are inverses to each other andinduce the bijection above.


3.3.7 For a map U ! V we have an isomorphism of sites S=U Š .S=V /=.U ! V /.Several formulas below implicitly contain this identification. For example, we have acanonical isomorphism

FjU Š .FjV /jU!Vfor F 2 ShS. For G;H 2 ShS this induces isomorphisms

HomSh S.G;H/jV Š HomSh S=V .GjV ;HjV /: (14)

3.3.8 We now consider a sheaf of abelian groups H 2 ShAb S. If � 2 covS.U / is acovering family of U andH 2 ShAbS, then we can define the Cech complex LC �.� IH/(see [BSS, 2.3.5]).

Definition 3.10. The sheaf H is called flabby if H i . LC �.� IH// Š 0 for all i 1,U 2 S and � 2 covS.U /.

For A 2 S we have the section functor

�.AI : : : / W ShAb S! Ab; �.AIH/ WD H.A/:This functor is left exact and admits a right-derived functor R�.AI : : : / which can becalculated using injective resolutions. Note that �.AI : : : / is acyclic on flabby sheaves,i.e. Ri�.AIH/ Š 0 for i 1 if H is flabby. Hence, the derived functor R�.AI : : : /can also be calculated using flabby resolutions. Note that an injective sheaf is flabby.

Lemma 3.11. The restriction functor ShAb S! ShAb S=Y preserves flabby sheaves.

Proof. Let H 2 ShAb S be flabby. For .U ! Y / 2 S=Y and � 2 covS.U / welet �jY 2 covS=Y .U ! Y / be the induced covering family as in 3.3.1. Note thatLC �.�jY IHjY / Š LC �.� IH/ is acyclic. Since covS=Y .U ! Y / is exhausted by families

of the form �jY , � 2 covS.U / it follows that HjY is flabby.

3.3.9 We now consider sheaves of abelian groups F;G 2 ShAb S.

Proposition 3.12. Assume that the site S has the property that for all U 2 S therestriction ShS ! ShS=U preserves representable sheaves. If Y 2 S, then inDC.ShAb S=Y / there is a natural isomorphism

RHomShAb S.F;G/jY Š RHomShAb S=Y .FjY ; GjY /:

Proof. We choose an injective resolution G ! I and furthermore an injective resolu-tion IjY ! J . Note that by Lemma 3.11 restriction is exact and therefore GjY ! J isa resolution of GjY . Then we have

RHomShAb S.F;G/jY Š HomShAb S.F; I /jY(14)Š HomShAb S=Y .FjY ; IjY /:

Furthermore,

RHomShAb S=Y .FjY ; GjY / Š HomShAb S=Y .FjY ; J /


and the map IjY ! J induces

RHomShAb S.F;G/jY Š HomShAb S=Y .FjY ; IjY /�! HomShAb S=Y .FjY ; J /Š RHomShAb S=Y .FjY ; GjY /:

(15)

If the restriction f � W ShS! ShS=Y would preserve injectives, then IjY ! J wouldbe a homotopy equivalence and the marked map would be a quasi-isomorphism.

In the generality of the present paper we do not know whether f � preserves injec-tives. Nevertheless we show that our assumption on S implies that the marked map in(15) is a quasi isomorphism for all sheaves F 2 ShAb S.

3.3.10 We first show a special case.

Lemma 3.13. IfF is representable, then the marked map in (15) is a quasi isomorphism.

Proof. Let .U ! Y / 2 S=Y and .A! Y / 2 S. Then on the one hand we have

HomShAb S=Y .Z.A/jY ; IjY /.U ! Y / D Hom.Sh S=Y /=.U!Y /..AjY /j.U!Y /; .IjY /jU!Y /(by 3.2.7)

Š HomSh S=U .AjU ; IjU /(by 3.3.7):

Since AjU is representable by our assumption on S there exists .B ! U/ 2 S=U suchthat AjU Š B ! U .

Since the restriction functor is exact (Lemma 3.8) and the restriction of an injectivesheaf from S to S=Y is still flabby (Lemma 3.11) we see that GjU ! IjU is a flabbyresolution. It follows that

HomSh S=U .AjU ; IjU / Š HomSh S=U .B ! U ; IjU /Š IjU .B ! U/ Š R�.B ! U;GjU /:

On the other hand

HomShAb S=Y .Z.A/jY ; J /.U ! Y / Š Hom.Sh S=Y /=.U!Y /..AjY /jU!Y ; JjU!Y /Š HomSh S=U .AjU ; JjU /Š HomSh S=U .B ! U ; JjU /Š J.B ! U/

Š R�.B ! U;GjU /:


3.3.11 We consider the functor which associates to F 2 ShAb S the cone

C.F / WD Cone�HomShAb S=Y .FjY ; IjY /! HomShAb S=Y .FjY ; J /

�:

In order to show that the marked map in (15) is a quasi isomorphism we must show thatC.F / is acyclic for all F 2 ShAb S.

The restriction functor .: : : /jY is exact by Lemma 3.8. For H 2 ShAb S=Y thefunctor HomShAb S=Y .: : : ;H/ is a right-adjoint. As a contravariant functor it transformscolimits into limits and is left exact. In particular, the functors

HomShAb S=Y ..: : : /jY ; IjY /; HomShAb S=Y ..: : : /jY ; J /

transform coproducts of sheaves into products of complexes. It follows thatF ! C.F /

also transforms coproducts of sheaves into products of complexes. If P 2 ShAb S is acoproduct of representable sheaves, then by Lemma 3.13 C.A/ is a product of acycliccomplexes and hence acyclic.

We claim that F ! C.F / transforms short exact sequences of sheaves to shortexact sequences of complexes. Since J is injective and .: : : /jY is exact, the functorF 7! HomShAb S=Y .FjY ; J / is a composition of exact functors and thus has this prop-erty. Furthermore we have HomShAb S=Y .FjY ; IjY / Š HomShAb S.F; I /jY . Since I isinjective, the functor F ! HomShAb S=Y .FjY ; IjY / has this property, too. This impliesthe claim.

We now argue by induction. Let n 2 N and assume that we have already shownthat H i .C.F // Š 0 for all F 2 ShAb S and i < n.

Consider F 2 ShAb S. Then there exists a coproduct of representables P 2 ShAb Sand an exact sequence

0! K ! P ! F ! 0:

In order to construct P observe that in general one has a canonical isomorphism

F Š colimA!FA:

We let P WDLA!F A. The collection of maps A! F in the index category induces

a canonical surjection

P DMA!F

A! colimA!FA Š F:

The short exact sequence of complexes

0! C.F /! C.P /! C.K/! 0

induces a long exact sequence in cohomology. Since H i .P / Š 0 for all i 2 Z weconclude that H i .C.F // Š H i�1.C.K// for all i 2 N. By our induction hypothesiswe get Hn.C.F // Š 0.

By induction on nwe show thatC.F / is acyclic for allF 2 ShAb S, and this impliesProposition 3.12.


3.3.12 Note that a site S which has finite products satisfies the assumption of Proposi-tion 3.12. In fact, for A 2 S we have by Lemma 3.6 that

AjY Š A � Y ! Y :

Therefore the restriction functor .: : : /jY preserves representables.

Corollary 3.14. Let S be as in Proposition 3.12. For sheavesF;G 2 ShAb S and Y 2 Swe have

ExtiShAb S.F;G/jY Š ExtiShAb S=Y .FjY ; GjY /:

Proof. Since .: : : /jY is exact (Lemma 3.8) we have

ExtiShAb S.F;G/jY Š .H iRHomShAb S.F;G//jYŠ H i .RHomShAb S.F;G/jY /Š H i .RHomShAb S=Y .FjY ; GjY // (by Proposition 3.12)

Š ExtiShAbS=Y .FjY ; GjY /:

3.3.13 Let us assume that S has finite fibre products. Fix U 2 S. Then we can definea morphism of sites U�W S! S=U which maps A 2 S to .U �A! U/ 2 S=U . Oneeasily checks the conditions given in [Tam94, 1.2.2]. This morphism of sites inducesan adjoint pair of functors

U�� W ShS, ShS=U W U �� :Let f W S=U ! S be the restriction defined in 3.3.2 (with Y replaced by U ) and let

f� W ShS=U , ShS W f �be the corresponding pair of adjoint functors.

Lemma 3.15. We assume that S has finite products. Then we have a canonical iso-morphism f � Š U��.Proof. We first define this isomorphism on representable sheaves. Since every sheafcan be written as a colimit of representable sheaves, U�� commutes with colimits asa left-adjoint, and f � commutes with colimits by Lemma 3.8, the isomorphism thenextends to all sheaves. Let W 2 S and F 2 ShS=U /. Then we have

HomSh S=U .U �� W ;F / Š HomSh S.W ;

U �� .F //Š U �� .F /.W /Š F.W � U ! U/

Š HomSh S=U ..W � U ! U/; F / (by Lemma 3.6)

Š HomSh S=U .f�W ;F /

This isomorphism is represented by a canonical isomorphismU �� .W / Š f �.W /

in ShS=U .


3.3.14 Let f W S! S0 be a morphism of sites. It induces an adjoint pair

f� W ShS, ShS0 W f �:Lemma 3.16. For F 2 ShS the functor f� has the explicit description

f�.F / WD colim .U!F /2Sh.S=F /f .U /:

Proof. In fact, for G 2 ShS0 we have

HomSh S0.colim .U!F /2Sh.S/=F f .U /;G/ Š lim .U!F /2Sh.S/=FHomSh S0.f .U /;G/

Š lim .U!F /2Sh.S/=FG.f .U //

Š lim .U!F /2Sh.S/=F f�G.U /

Š lim .U!F /2Sh.S/=FHomSh S0.U ; f �G/Š HomSh S.colim .U!F /2Sh.S/=FU ; f

�G/Š HomSh S.F; f

�G/:

Lemma 3.17. For A 2 S we have

f�.A/ D f .A/:Proof. Since .idA W A! A/ 2 S=A is final we have

f�.A/ Š colim .U!A/2Sh.S/=Af .U /

Š colim .U!A/2S=Af .U /

Š f .A/:Lemma 3.18. If f W S! S0 is fully faithful, then we have forA 2 S that f �f�A Š A.

Proof. We calculate for U 2 S that

f �f�A.U / Š f �f .A/.U / (by Lemma 3.17)

Š f .A/.f .U //Š HomS0.f .U /; f .A//

Š HomS.U;A/

Š A.U /3.3.15 For U 2 S we have an induced morphism of sites Uf W S=U ! S0=f .U /which maps .A! U/ to .f .A/! f .U //.

If f is fully faithful, then Uf is fully faithful for all U 2 S.

Lemma 3.19. For G 2 ShS0 we have in S=U the identity

Uf�.Gjf .U// Š .f �G/jU :

If we know in addition that S, S0 have finite products which are preserved by f W S! S0,then for F 2 S we have in S0=f .U / the identity

Uf�FjU Š .f�F /jf .U/:


Proof. Indeed, for .V ! U/ 2 S=U we have

.f �G/jU .V ! U/ D G.f .V //D Gjf .U/.f .V /! f .U // D .Uf �Gjf .U//.V ! U/:

In order to see the second identity note that we can write

FjU Š colim .A!FjU /2Sh.S=U /=F

jUA:

Since Uf� is a left-adjoint it commutes with colimits. Furthermore the restrictionfunctors .: : : /jU and .: : : /f .U/ are exact by Lemma 3.8 and therefore also commutewith colimits. Writing

F Š colim .A!F /2Sh.S/=FA

we get

FjU Š .colim .A!F /2Sh.S/=FA/jU Š colim .A!F /2Sh.S/=FAjUŠ colim .A!F /2Sh.S/=FA � U ! A (by Lemma 3.6).

Using that f preserves products in the isomorphism marked by .Š/ we calculate

Uf�FjU Š Uf�colim .A!F /2Sh.S/=FA � U ! U

Š colim .A!F /2Sh.S/=F Uf�A � U ! U

Š colim .A!F /2Sh.S/=F f .A � U/! f .U / (by Lemma 3.17)

.Š/Š colim .A!F /2Sh.S/=F f .A/ � f .U /! f .U /

Š colim .A!F /2Sh.S/=F f .A/jf .U/Š .colim .A!F /2Sh.S/=F f .A//jf .U/Š .f�F /jf .U/: (by Lemma 3.16)

Lemma 3.20. Assume that S;S0 have finite products which are preserved byf W S! S0.For F 2 ShS and G 2 ShS0 we have a natural isomorphism

HomSh S.F; f�G/ Š f �HomSh S0.f�F;G/:

Proof. For U 2 S we calculate

HomSh S.F; f�G/.U / D HomSh S=U .FjU ; .f �G/jU /

Š HomSh S=U .FjU ; Uf �.Gjf .U/// (by Lemma 3.19)

Š HomSh S0=f .U /.Uf�.FjU /; Gjf .U//

Š HomSh S0=f .U /..f�F /jf .U/; Gjf .U// (by Lemma 3.19)

D HomSh S0.f�F;G/.f .U //D f �HomSh S0.f�F;G/.U /


3.3.16 We now consider the derived version of Lemma 3.20. Let F 2 ShAb S andG 2 ShAb S0 be sheaves of abelian groups.

Proposition 3.21. We make the following assumptions:

1. The sites S;S0 have finite products.

2. The morphism of sites f W S! S0 preserves finite products.

3. We assume that f � W ShS0 ! ShS is exact.

Then in DC.ShAb S/ there is a natural isomorphism

RHomShAb S.F; f�G/ Š f �RHomShAb S0.f�F;G/:

Proof. By [Tam94, Proposition 3.6.7] the conditions on the sites and f imply that thefunctor f� is exact. It follows that f � preserves injectives (see e.g. [Tam94, Proposi-tion 3.6.2]). We choose an injective resolution G ! I �. Then f �G ! f �I � is aninjective resolution of f �G. It follows that

f �RHomShAb S0.f�F;G/ Š f �HomShAb S0.f�F; I �

/

Š HomShAb S.F; f�I �

/ Š RHomShAb S.F; f�G/:

Proposition 3.21 is very similar in spirit to Proposition 3.12. On the other hand,we can not deduce 3.12 from 3.21 since the functor f W ShS=U ! S in general doesnot preserve products. In fact, if S has fibre products, then .A ! U/ � .B ! U/ D.A �U B ! U/. Then f ..A ! U/ � .B ! U// Š A �U B while f .A !U/ � f .B ! U/ Š A � B .

3.3.17 Recall the notations W�;W� from 3.3.13.

Lemma 3.22. Let W 2 S. For every F 2 ShS and A 2 S we have

HomSh S.W ; F /.A/ Š W �� W �� F.A/:Proof. For A 2 S we consider f W S=A! S and get

HomSh S.W ; F /.A/ Š HomSh S=A.W jA; FjA/Š HomSh S=A.f

�W ; f �F /Š HomSh S=A.

A �� .W /; f �F / (by Lemma 3.15)

Š HomSh S.W ;A �� f �.F //

Š A �� .f �F /.W / (16)

Š f �F.A �W ! A/

Š F.A �W /Š W �� .f �F /.A/Š W �� .W �� .F //.A/ (by Lemma 3.15):


3.3.18 To abbreviate, let us introduce the following notation.

Definition 3.23. If S has finite products, then for W 2 S we introduce the functor

RW WD W �� ıW�� W ShS! ShS:

We denote its restriction to the category of sheaves of abelian groups by the samesymbol. Since W�� is exact and W�� is left-exact, it admits a right-derived functorRRW W DC.ShAb S/! DC.ShAb S/.

Lemma 3.24. Let F 2 ShAb S and W 2 S. Then we have a canonical isomorphismRHomShAb S.Z.W /; F / Š RRW .F /.

Proof. This follows from the isomorphism of functors HomShAb S.Z.W /; : : : / ŠRW .: : : / from ShAb S to ShAb S. In fact, we have for F 2 ShAb S that

HomShAb S.Z.W /; F / Š HomSh S.W ;F .F //

Š W �� .W �� .F // (Equation 16) (17)

Š RW .F / (Definition 3.23):

3.4 Application to sites of topological spaces

3.4.1 In this subsection we consider the site S of compactly generated topologicalspaces as in 3.1.2 and some of its sub-sites. We are interested in proving that restrictionto sub-sites preserve Exti -sheaves.

We will further study properties of the functor RW . In particular, we are interestedin results asserting that the higher derived functors RiRW .F /, i 1 vanish undercertain conditions on F and W .

Lemma 3.25. If C 2 S is compact and H is a discrete space, then Map.C;H/ isdiscrete, and

RC .H/ D Map.C;H/:

Proof. We first show that Map.C;H/ is a discrete space in the compact-open topology.Let f 2 Map.C;H/. Since C is compact, the image f .C / is compact, hence finite.We must show that ¹f º� Map.C;H/ is open. Let h1; : : : ; hr be the finite set of valuesof f . The sets f �1.hi / � C are closed and therefore compact and their union is C .The sets ¹hiº� H are open. Therefore Ui WD ¹g 2 Map.C;H/ j g.f �1.hi // � ¹hiººare open subsets of Map.C;H/. We now see that ¹f ºDTr

iD1 Ui is open.We have by the exponential law

RC .H/.A/ Š H.A � C/ Š HomS.A � C;H/Š HomS.A;Map.C;H// Š Map.C;H/.A/:


3.4.2 A sheaf G 2 ShAb S is called RW -acyclic if RiRW .G/ Š 0 for i 1.

Lemma 3.26. If G is a discrete group, then G is RRn-acyclic.

Proof. Let G ! I � be an injective resolution. Then we have for A 2 S that

RRn.I�

/.A/(16)Š I

�

.A �Rn/:

Therefore RiRRn.G/ is the sheafification of the presheaf

A 7! H i .I�

.A �Rn//:

Since G is discrete, the sheaf cohomology of G is homotopy invariant, and therefore

H i .I�

.A �Rn// Š H i .A �RnIG/ ŠŠ H i .AIG/:To be precise this can be seen as follows. Let .A/ denote the site of open subsetsof A. It comes with a natural map A W .A/ ! S. The sheaf cohomology functor isthe derived functor of the evaluation functor. In order to indicate on which categorythis evaluation functor is defined we temporarily use subscripts. If I 2 ShAb S isinjective, then �AI 2 ShAb.A/ is still flabby (see [BSS, Lemma 2.32]). This impliesthat H�ShAb S.AIG/ Š H�.A/.A; �A.G//.

The diagram

AidA�0 ��

idA

��

��

� A �Rn

prA��

A

is a homotopical isomorphism (in the sense of [KS94, 2.7.4])A! A�Rn overA. Wenow apply [KS94, 2.7.7] which says that the natural map

R.prA/�pr�A.�A.G//! R.idA/�id�A.�A.G//

is an isomorphism in DC.ShAb.A//. But since G is discrete we get pr�A.�AG/ Š�A�Rn.G/. If we apply the functor R�.A/.A; : : : / to this isomorphism and take coho-mology we get the desired isomorphism marked by Š.4

Now, the sheafification of the presheaf S 3 A! H i .AIG/ 2 Ab is exactly the i thcohomology sheaf of I � which vanishes for i 1.

Lemma 3.27. The sheaf Rn is RW -acyclic for every compact W 2 S.

Proof. Let Rn ! I � be an injective resolution. Then RiRW .Rn/ is the sheafification

of the presheaf

S 3 A 7! H i .HomShAb S.Z.W jA/; I�

jA// Š H i .I�

.A �W //:4It is tempting to apply a Künneth formula to calculateH�.A�Rn;G/. But sinceA is not necessarily

compact it is not clear that the Künneth formula holds.


Let Œs� 2 H i .I �.A�W // be represented by s 2 I i .A�W /, and a 2 A. Then we mustfind a neighbourhood U � A of a such that Œs�jU�W D 0, i.e. sjU�W D dt for somet 2 I i�1.U �W /, where d W I i�1 ! I i is the boundary operator of the resolution.

The ring structure of R induces on Rn the structure of a sheaf of rings. In order todistinguish this sheaf of rings from the sheaf of groups Rn we will use the notation C .Note that Rn is in fact a sheaf of C -modules.

The forgetful functor res W ShC -mod S! ShAb S fits into an adjoint pair

ind W ShAb S, ShC -mod S W res;

where ind is given by ShAb S 3 V ! V ˝Z C 2 ShC -mod. Since C is a torsion-freesheaf it is flat. It follows that ind is exact and res preserves injectives.

We can now choose an injective resolution Rn ! J � in ShC -mod S and assume thatI � D res.J �/.

Since the complex of sheaves I � is exact we can find an open covering .Vr/r2R ofA � W such that sjVr D dtr for some tr 2 I i�1.Vr/. Since W is compact (locallycompact suffices), by [Ste67, Theorem 4.3]) the product topology on A � W is thecompactly generated topology used for the product in S. Hence after refining thecovering .Vr/ we can assume that Vr D Ar � Wr for open subsets Ar � A andWr � W for all r 2 R.

We define Ra WD ¹r 2 R j a 2 Arº. The family .Wr/r2Ra is an open coveringof W . Since W is compact we can choose a finite set r1; : : : ; rk 2 Ra such thatW WD .Wr1 ; : : : ; Wrk / is still an open covering of W . The subset U WD Tk

jD1Arj isan open neighbourhood of a 2 A.

Since I i�1 is injective we can choose5 extensions Qtr 2 I i�1.A � W / such that.Qtr/jVr D tr .

We choose a partition of unity .�1; : : : ; �v/ subordinate to the finite covering W .We take advantage of the fact that I � D res.J �/ which implies that we can multiplysections by continuous functions, and that d commutes with this multiplication. Wedefine

t WDvXkD1

�k.Qtrk /jU�W 2 I i�1.U �W /:

Note that �k.s � d Qtrk /jU�W D 0. In fact we have �k.s � d Qtrk /j.U�W /\Vrk D �k.s �dtrk /j.U�W /\Vrk D 0. Furthermore, there is a neighbourhoodZ of the complement of.U �W /\ Vrk in U �W where �k vanishes. Therefore the restrictions �k.s � d Qtrk /vanish on the open covering ¹Z; .U � W / \ Vrk º of U � W , and this implies the

5Let U � X be an open subset. Then we have an injection U ! X and hence an injection Z.U /!Z.X/. For an injective sheaf I we get a surjection HomShAb S.Z.X/; I /! HomShAb S.Z.U /; I /. In othersymbols, I.X/! I.U / is surjective.


assertion. We get

dt DvX

jD1d.�k.Qtrk /jU�W / D

vXjD1

�kd.Qtrk /jU�W

DvX

jD1�k.d Qtrk /jU�W D

vXjD1

�ksjU�W

D sjU�W :Corollary 3.28. 1. If G is discrete, then we have ExtiShAb S.Z.R

n/; G/ Š 0 for alli 1.

2. For every compact W 2 S and n 1 we have ExtiShAb S.Z.W /;Rn/ Š 0 for all

i 1.

Lemma 3.29. If W is a profinite space, then every sheaf F 2 ShAb S is RW -acyclic.Consequently, ExtiShAb S.Z.W /; F / Š 0 for i 1.

Proof. We first show the following intermediate result which is used in 3.4.3 in orderto finish the proof of Lemma 3.29.

Lemma 3.30. If W is a profinite space, then �.W I : : : / is exact.

Proof. A profinite topological space can be written as limit,W Š limIWn, for an inversesystem of finite spaces .Wn/n2I . Let pn W W ! Wn denote the projections. First of all,W is compact. Every covering ofW admits a finite subcovering. Furthermore, a finitecovering admits a refinement to a covering by pairwise disjoint open subsets of the form¹p�1n .x/ºx2Wn for an appropriate n 2 I . This implies the vanishing LHp.W; F / Š 0

of the Cech cohomology groups for p 1 and every presheaf F 2 PrAb S.Let Hq D Rqi be the derived functor of the embedding i W ShAb S ! PrAb S of

sheaves into presheaves. We now consider the Cech cohomology spectral sequence[Tam94, 3.4.4] .Er ; dr/ ) R��.W;F / with Ep;q2 Š LHp.W IHq.F // and use[Tam94, 3.4.3] to the effect that LH 0.W IHq.F // Š 0 for all q 1. Combiningthese two vanishing results we see that the only non-trivial term of the second page ofthe spectral sequence isE0;02 Š LH 0.W IH0.F // Š F.W /. Vanishing ofRi�.W I : : : /for i 1 is equivalent to the exactness of �.W I : : : /.3.4.3 We now prove Lemma 3.29. Let i 1. We use that RiRW .F / is the sheafifi-cation of the presheaf S 3 A 7! Ri�.A �W IF / 2 Ab. For every sheaf F 2 ShAb Swe have by some intermediate steps in (16)

�.A �W IF / Š �.W IRA.F //:

Let us choose an injective resolution F ! I �. Using Lemma 3.30 for the secondisomorphism we get

H i�.AIRW .I�

//s 7!QsŠ H i�.A �W I I �

/Qs 7!NsŠ �.W IH iRA.I

�

//: (18)


Consider a point a 2 A and s 2 H i�.AIRW .I�//. We must show that there exists

a neighbourhood a 2 U � A of a such that sjU D 0. Since I � is an exact sequenceof sheaves in degree 1 there exists an open covering ¹Yrºr2R of A � W such thatQsjYr D 0. After refining this covering we can assume thatYr Š Ur�Vr for suitable opensubsets Ur � A and Vr � W . Consider the set Ra WD ¹r 2 R j a 2 Urº. Since W iscompact the covering ¹Vrºr2Ra ofW admits a finite subcovering indexed by Z � Ra.The set U WDT

r2Z Ur � A is an open neighbourhood of a. By further restriction weget QsjU�Vr D 0 for all r 2 Z. By (18) this means that 0 D sjU jVr 2 �.W IH iRU .I

�//.Therefore sjU vanishes locally onW and therefore globally. This implies sjU D 0.

3.4.4 Let f W Slc ! S be the inclusion of the full subcategory of locally compacttopological spaces. Since an open subset of a locally compact space is again locallycompact we can define the topology on Slc by

covSlc.A/ WD covS.A/; A 2 Slc:

The compactly generated topology and the product topology on products of locallycompact spaces coincides. The same applies to fibre products. Furthermore, a fibreproduct of locally compact spaces is locally compact. The functor f preserves fibreproducts. In view of the definition of the topology Slc the inclusion functor f W Slc ! Sis a morphism of sites.

Lemma 3.31. Restriction to the site Slc commutes with sheafification, i.e.

i] ı pf � Š f � ı i]:Proof. (Compare with the proof of Lemma 3.7.) This follows from covSlc.A/ ŠcovS.A/ for all A 2 S and the explicit construction of i] in terms of the set covSlc

(see [Tam94, Section 3.1]).

Lemma 3.32. The restriction f � W ShS! ShSlc is exact.

Proof. (Compare with the proof of Lemma 3.8.) The functor f � is a right-adjoint andthus commutes with limits. Colimits of presheaves are defined object-wise, i.e. for adiagram

C ! Pr S; c 7! Fc

of presheaves we have

. pcolimc2CFc/.U / D colimc2CFc.U /:

It follows from the explicit description of pf � that this functor commutes with colimitsof presheaves. For a diagram of sheaves F W C ! ShS we have

colimc2CFc D i] pcolimc2CFc :

By Lemma 3.31 we getf �colimc2CFc Š f �i] pcolimc2CFc Š i] pf � pcolimc2CFc Ši] pcolimc2C

pf �Fc Š colimc2Cf�Fc :


3.4.5 We have now verified that f W Slc ! S satisfies the assumptions of Proposi-tion 3.21.

Corollary 3.33. Let f W Slc ! S be the inclusion of the site of locally compact spaces.For F 2 ShAb Slc and G 2 ShAb S we have

f �RHomShAb S.f�F;G/ Š RHomShAb Slc.F; f �G/:

In particular we have

f �ExtkShAb S.f�F;G/ Š ExtkShAb Slc.F; f �G/

for all k 0.

In fact, the first assertion implies the second since f � is exact.We need this result in the following special case. IfG 2 S is a locally compact group,

then by abuse of notation we write G for the sheaves of abelian groups represented byG in both categories ShAb S and ShAb Slc.

We have f �G D G. By Lemma 3.17 we also have f�G Š G.

Corollary 3.34. Let G;H 2 Slc be locally compact abelian groups. We have

f �ExtkShAb S.G;H/ Š ExtkShAb Slc.G;H/

for all k 0.

3.4.6 In some places we will need a second sub-site of S, the site Sloc-acyc of locallyacyclic spaces.

Definition 3.35. A space U 2 S is called acyclic, if H i .U IH/ Š 0 for all discreteabelian groups H and i 1.

By Lemma 3.30 all profinite spaces are acyclic. The space Rn is another exampleof an acyclic space. In fact, the homotopy invariance used in the proof of 3.26 showsthat the inclusion 0! Rn induces an isomorphism H i .RnIH/ Š H i .¹0ºIH/, and aone-point space is clearly acyclic.

Definition 3.36. A space A 2 S is called locally acyclic if it admits an open coveringby acyclic spaces.

In general we do not know if the product of two locally acyclic spaces is againlocally acyclic (the Künneth formula needs a compactness assumption). In order toensure the existence of finite products we consider the combination of the conditionslocally acyclic and locally compact.

Note that all finite-dimensional manifolds are locally acyclic and locally compact.An open subset of a locally acyclic locally compact space is again locally acyclic. Welet Slc-acyc � S be the full subcategory of locally acyclic locally compact spaces. Thetopology on Slc-acyc is given by

covSlc-acyc.A/ WD covS.A/:

Let g W Slc-acyc ! S be the inclusion. The proofs of Lemma 3.31, Lemma 3.32 andCorollary 3.33 apply verbatim.


Corollary 3.37. 1. The restriction g� W ShS! ShSlc-acyc is exact.2. For F 2 ShAb Slc-acyc and G 2 ShAb S we have

g�RHomShAb S.g�F;G/ Š RHomShAb Slc-acyc.F; g�G/

and

g�ExtkShAb S.g�F;G/ Š ExtkShAb Slc-acyc.F; g�G/

for all k 0.

Corollary 3.38. Let G;H 2 Slc-acyc be locally acyclic abelian groups. We have

g�ExtkShAb S.G;H/ Š ExtkShAb Slc-acyc.G;H/

for all k 0.

Note that the productQ

N T is compact but not locally acyclic.

3.5 Zmult-modules

3.5.1 We consider the multiplicative semigroup Zmult of non-zero integers. Everyabelian group G (written multiplicatively) has a tautological action of Zmult by homo-morphisms given by

Zmult �G ! G; .n; g/ 7! gn:

Definition 3.39. When we consider G with this action we write G.1/.

The semigroup Zmult will therefore act on all objects naturally constructed from anabelian group G. We will in particular use the action of Zmult on the group-homologyand group-cohomology of G.

If G is a topological group, then Zmult acts by continuous maps. It therefore alsoacts on the cohomology of G as a topological space. Also this action will play a rolelater.

The remainder of the present subsection sets up some language related to Zmult-actions.

Definition 3.40. A Zmult-module is an abelian group with an action of Zmult by homo-morphisms.

We will write this action as Zmult � G 3 .m; g/ 7! ‰m.g/ 2 G. Thus in the caseof G.1/ we have ‰m.g/ D gm.

3.5.2 We let Zmult-mod denote the category of Zmult-modules. An equivalent descrip-tion of this category is as the category of modules under the commutative semigroupring ZŒZmult�. The category Zmult-mod is an abelian tensor category.

We have an exact inclusion of categories

Ab ,! .Zmult-mod/; G 7! G.1/


By

F W .Zmult-mod/! Ab

we denote the forgetful functor.

Definition 3.41. LetG be an abelian group. Fork 2 Z we letG.k/ 2 Zmult-mod denotethe Zmult-module given by the action Zmult �G 3 .p; g/ 7! ‰p.g/ WD gpk 2 G.

Observe that for abelian groups V;W we have a natural isomorphism

V.k/˝Z W.l/ Š .V ˝Z W /.k C l/: (19)

3.5.3 Let V be a Zmult-module.

Definition 3.42. We say that V has weight k if there exists an isomorphism V ŠF .V /.k/ of Zmult-modules.

If V has weight k, then every sub-quotient of V has weight k. Note that aZmult-module can have many weights. We have e.g. isomorphisms of Zmult-modules.Z=2Z/.1/ Š .Z=2Z/.k/ for all k 6D 0.

3.5.4 Let V 2 Zmult-mod. We say that v 2 V has weight k if it generates a submoduleZ < v >� V of weight k. For k 2 N we let Wk W .Zmult-mod/ ! Ab be the functorwhich associates to V 2 Zmult-mod its subgroup of vectors of weight k. Then we havean adjoint pair of functors

.k/ W Ab, .Zmult-mod/ W Wk :

Observe that the functorWk is not exact. Consider for example a prime number p 2 Nand the sequence

0! Z.1/p! Z.1/! .Z=pZ/.p/! 0:

The projection map is indeed Zmult-equivariant since mp � m mod p for all m 2 Z.Then

0 Š Wp.Z.1//! Wp..Z=pZ/.p// Š Z=pZ

is not surjective.

3.5.5 Let V 2 Ab and V.1/ 2 Zmult-mod. Then we can form the graded tensor algebra

T �Z.V .1// D Z˚ V.1/˚ V.1/˝Z V.1/˚ :

We see that T k.V .1// has weight k. The elements x ˝ x 2 V.1/˝ V.1/ generate ahomogeneous ideal I . Hence the graded algebra ƒ�Z.V .1// WD T �Z.V .1//=I has theproperty that ƒkZ.V .1// has weight k.


3.5.6 It makes sense to speak of a sheaf or presheaf of Zmult-modules on the site S.We let ShZmult-modS and PrZmult-mod S denote the corresponding abelian categories ofsheaves and presheaves.

Definition 3.43. Let V 2 ShZmult-modS (or V 2 PrZmult-mod S) and k 2 Z. We say that

V is of weight k if the map Vmk�m��! V vanishes for all m 2 Zmult.

We define the functors .k/ W ShAb S ! ShZmult-modS, F W ShZmult-modS ! ShAb S,and Wk W ShZmult-modS ! ShAb S (and their presheaf versions) object-wise. We alsohave a pair of adjoint functors

.k/ W ShAb S, ShZmult-modS W Wk(and the corresponding presheaf version). A sheaf V 2 ShZmult-modS of Zmult-moduleshas weight k 2 Z if V Š F .V /.k/ Š Wk.V /.k/.

4 Admissibility of sheaves represented by topological abeliangroups

4.1 Admissible sheaves and groups

4.1.1 The main topic of the present paper is a duality theory for abelian group stacks(Picard stacks, see 2.4) on the site S. A Picard stack P 2 PIC.S/ gives rise to thesheaf of objectsH 0.P / and the sheaf of automorphisms of the neutral objectH�1.P /.These are sheaves of abelian groups on S.

We will define the notion of a dual Picard stackD.P / (see 5.4). With the intentionto generalize the Pontrjagin duality for locally compact abelian groups to group stackswe study the question under which conditions the natural map P ! D.D.P // is anisomorphism. In Theorem 5.11 we see that this is the case if the sheaves are dualizable(see 5.2) and admissible (see 4.1). Dualizability is a sheaf-theoretic generalization ofthe classical Pontrjagin duality and is satisfied e.g. for the sheavesG for locally compactgroups G 2 S (see 5.3). Admissibility is more exotic and will be defined below (4.1).One of the main results of the present paper asserts that the sheaves G are admissiblefor a large (but not exhaustive) class of locally compact groups G 2 S, and for an evenlarger class if S is replaced by Slc-acyc.

The main tool in our computations is a certain double complex, which will beintroduced in Section 4.2. Similar techniques were used by Lawrence Breen in [Bre69],[Bre76], [Bre78].

Definition 4.1. We call a sheaf of groups F admissible if ExtiShAbS.F;T / Š 0 fori D 1; 2. A topological abelian group G 2 S is called admissible if G is an admissiblesheaf.

We will also consider the sub-sites Slc-acyc � Slc � S of locally compact spaces.


Definition 4.2. A locally compact abelian topological group G 2 Slc is called admis-sible on Slc if ExtiShAbSlc

.G;T / Š 0 for i D 1; 2 (and correspondingly for Slc-acyc).

Lemma 4.3. Let f W Slc ! S be the inclusion. If F 2 ShAb Slc and f�F is admissible,then F is admissible on Slc.

Proof. This is an application of Corollary 3.33.

Corollary 4.4. If G 2 Slc is admissible, then it is admissible on Slc.

This is an application of Corollary 3.34.

Lemma 4.5. The class of admissible sheaves is closed under finite products and ex-tensions.

Proof. For finite products the assertions follows from the fact that ExtShAb S.: : : ;T /commutes with finite products. Given an extension of sheaves

0! F ! G ! H ! 0

such that F andH are admissible, then alsoG is admissible. This follows immediatelyfrom the long exact sequence obtained by applying Ext�ShAb S.: : : ;T /.

4.1.2 In this paragraph we formulate one of the main theorems of the present paper.Let G be a locally compact topological abelian group. We first recall some technicalconditions.

Definition 4.6. We say that G satisfies the two-three condition, if

1. it does not admitQn2N Z=2Z as a subquotient,

2. the multiplication by 3 on the component G0 of the identity has finite cokernel.

Definition 4.7. We say thatG is locally topologically divisible if for all primes p 2 Nthe multiplication map p W G ! G has a continuous local section.

Using that the class of admissible groups is closed under finite products and exten-sions we get the following general theorem.

Theorem 4.8. 1. If G is a locally compact abelian group which satisfies the two-threecondition, then it is admissible over Slc-acyc.

2. Assume that

(a) G satisfies the two-three condition,

(b) G has an open subgroup of the form C �Rn with C compact such thatG=C �Rn

is finitely generated

(c) the connected component of the identity of G is locally topologically divisible.

Then G is admissible over Slc.


Proof. By [HM98, Theorem 7.57(i)] the groupG has a splittingG Š H �Rn for somen 2 N0, whereH has a compact open subgroupU . The quotientG=.U �Rn/ Š H=Uis therefore discrete. Using Lemma 4.5 conclude that G is admissible over Slc if Rn

and H are so.Admissibility of Rn follows from Theorem 4.33 (which we prove later) in conjunc-

tion with 4.5 and 4.4.Since D WD H=U is discrete, the exact sequence

0! U ! H ! H=U ! 0

has local sections and therefore induces an exact sequence of associated sheaves

0! U ! H ! D ! 0

by Lemma 3.4. If D is finitely generated, then it is admissible by Theorem 4.18,and hence admissible on Slc by Lemma 4.4. Otherwise it is admissible on Slc-acyc byTheorem 4.28.

Therefore H is admissible on Slc (or Slc-acyc, respectively) by Lemma 4.5 if U isadmissible over Slc (or Slc-acyc, respectively). The compact group U fits into an exactsequence

0! U0 ! U ! P ! 0

whereP is profinite andU0 is closed and connected. By assumptionU ,U0 andP satisfythe two-three condition. The connected compact group U0 is admissible on Slc (or onSlc without the assumption that it is locally topologically divisible), by Theorem 4.79,and the profinite P is admissible by Theorem 4.66, and hence admissible on Slc byLemma 4.4.

Note that0! U 0 ! U ! P ! 0

is exact by Lemma 4.39. Now it follows from Lemma 4.5 that U is admissible on Slc

(or Slc-acyc, respectively).

4.1.3 We conjecture that the assumption thatG satisfies the two-three condition is onlytechnical and forced by our technique to prove admissibility of compact groups.

In the remainder of this section, we will mainly be concerned with the proof of thestatements used in the proof of Theorem 4.8 above.

4.1.4 Our proofs of admissibility for a sheaf F 2 ShAb S will usually be based on thefollowing argument.

Lemma 4.9. Assume that F 2 ShAb S satisfies

1. ExtiShAb S.F;Z/ Š 0 for i D 2; 3;

2. ExtiShAb S.F;R/ Š 0 for i D 1; 2.

Then F is admissible.


Proof. We apply the functor Ext�ShAb S.F; : : : / to the sequence

0! Z! R! T ! 0

and get the following segments of the long exact sequence

! ExtiShAb S.F;R/! ExtiShAb S.F;T /! ExtiC1ShAb S.F;Z/! :We see that the assumptions on F imply that ExtiShAb S.F;T / Š 0 for i D 1; 2.

4.1.5 A space W 2 S is called profinite if it can be written as the limit of an inversesystem of finite spaces. Lemma 3.29 has as a special case the following theorem.

Theorem 4.10. If W 2 S is profinite, then Z.W / is admissible.

4.2 A double complex

4.2.1 Let H 2 ShAb S be a sheaf of groups with underlying sheaf of sets F .H/ 2ShS. Applying the linearization functor Z (see 3.2.6) we get again a sheaf of groupsZ.F .H// 2 ShAb S. The group structure of H induces on Z.F .H// a ring structure.We denote this sheaf of rings by ZŒH �. It is the sheafification of the presheaf pZŒH �which associates to A 2 S the integral group ring pZŒH �.A/ of the group H.A/.

We consider the category ShZŒH�-modS of sheaves of ZŒH �-modules. The trivialaction of the sheaf of groups H on the sheaf Z induces the structure of a sheaf ofZŒH �-modules on Z. With the notation introduced below we could (but refrain fromdoing this) write this sheaf of ZŒH �-modules as coind.Z/.

4.2.2 The forgetful functor res W ShZŒH�-modS ! ShAb S fits into an adjoint pair offunctors

ind W ShAb S, ShZŒH�-modS W res:

Explicitly, the functor ind is given by

ind.V / WD ZŒH �˝Z V:

Since ZŒH � is a torsion-free sheaf and therefore a sheaf of flat Z-modules the functorind is exact. Consequently the functor res preserves injectives.

4.2.3 We let coinv W ShZŒH�-modS! ShAb S denote the coinvariants functor given by

ShZŒH�-modS 3 V 7! coinv.V / WD V ˝ZŒH� Z 2 ShAb S:

This functor fits into the adjoint pair

coinv W ShZŒH�-modS, ShAb S W coind

with the coinduction functor which maps W 2 ShAb S to the sheaf of ZŒH �-modulesinduced by the trivial action of H in W , formally this can be written as

coind.W / WD HomShAbS.Z; W /:


Lemma 4.11. Every sheaf F 2 ShZŒH�-modS is a quotient of a flat sheaf.

Proof. Indeed, the counits of the adjoint pairs .Z.: : : /;F / and .ind; res/ induce asurjection ind.Z.F .res.F //// ! F . Explicitly it is given by the composition of thesum and action map (omitting to write some forgetful functors)

ZŒH �˝Z ZŒF �! ZŒH �˝ F ! F:

Since ZŒH � is a sheaf of unital rings this action is surjective. Moreover, for A 2ShZŒH�-modS we have

A˝ZŒH� .ZŒH �˝Z ZŒF �/ Š A˝Z ZŒF �:

Since ZŒF � is a torsion-free sheaf of abelian groups the operation A ! A ˝ZŒH�

.ZŒH �˝Z ZŒF �/ preserves exact sequences in ShZŒH�-modS. Therefore ZŒH �˝Z ZŒF �is a flat sheaf of ZŒH �-modules.

Lemma 4.12. The class of flat ZŒH �-modules is coinv-acyclic.

Proof. Let F � be an exact lower bounded homological complex of flat ZŒH �-modules.We choose a flat resolution P � ! Z in ShZŒH�-modS which exists by Lemma 4.116.Since F � consists of flat modules the induced map

F� ˝ZŒH� P

� ! F� ˝ZŒH� Z D coinv.F �

/

is a quasi-isomorphism. Since P � consists of flat modules, tensoring by P � commuteswith taking cohomology, so that we have

H�.F � ˝ZŒH� P�

/ Š H�.H�.F �

/˝ZŒH� P�

/ Š 0:Therefore coinv.: : : / maps acyclic complexes of flat ZŒH �-modules to acyclic com-plexes of Z-modules.

Corollary 4.13. We can calculate L�coinv.Z/ using a flat resolution.

4.2.4 We will actually work with a very special flat resolution of Z. The bar construc-tion on the sheaf of groups H gives a sheaf H � of simplicial sets with an action of H .We let C �.H/ WD C.Z.H �// be the sheaf of homological chain complexes associatedto the sheaf of simplicial groups Z.H �/. TheH -action onH � induces anH -action onC �.H/ and therefore the structure of a sheaf of ZŒH �-modules. In order to understandthe structure ofC �.H/we first consider the presheaf version pC �.H/ WD C. pZ.H �//.In fact, we can write

H i Š H �H � �H„ ƒ‚ …i factors

;

as a sheaf of H -sets, and therefore

pC i .H/ Š pZŒH �˝pZ pZ.H � �H„ ƒ‚ …i factors

/ (20)

6Note that P � is a homological complex, i.e. the differentials have degree �1.


The cohomology of the complex pC �.H/ is given by

H i . pC�

.H// Š´pZ; i D 0;0; i 1;

where pZ 2 PrAb S denotes the constant presheaf with value Z. Since sheafification i]

is an exact functor and by definition C �.H/ D i] pC �.H/ we get

H i .C�

.H// Š´

Z; i D 0;0; i 1:

Furthermore,

C i .H/ Š ZŒH �˝Z Z.H � �H„ ƒ‚ …i factors

/ D ind.Z.H � �H„ ƒ‚ …i factors

// (21)

shows that C i .H/ is flat. Let us write C � WD C �.H/ D ind.D�/ with

Di WD Z.H � �H„ ƒ‚ …i factors

/:

Definition 4.14. For a sheaf H 2 ShAb S we define the complex U � WD U �.H/ WDcoinv.C �/.

It follows from the construction that U � depends functorially on H . In particular,for ˛ W H ! H 0 we have a map of complexes U �.˛/ W U �.H/! U �.H 0/.

4.2.5 The main tool in our proofs of admissibility of a sheaf F is the study of thesheaves

R�HomShZŒH�-modS.Z; coind.W //

for W D Z;R. Let us write this in a more complicated way using the special flatresolution C �.H/ ! Z constructed in 4.2.4. We choose an injective resolutioncoind.W / ! I � in ShZŒH�-modS. Using that res ı coind D id, and that res.I �/ isinjective in ShAb S, we get7

RHomShZŒH�-modS.Z; coind.W // Š HomShZŒH�-modS.Z; I�

/

Š HomShZŒH�-modS.C�

.H/; I�

/

Š HomShZŒH�-modS.ind.D�

/; I�

/

Š HomShAb S.D�

; res.I �

//

D HomShAb S.D�

; res.coind.res.I �

////

D HomShZŒH�-modS.ind.D�

/; coind.res.I �

///

7Note thatD� is considered in this calculation as a sequence of sheaves, not as a complex. The differentialsin the intermediate steps involving D� are still induced via the isomorphisms from the differentials of C �

and I �.


Š HomShZŒH�-modS.C�

; coind.res.I �

///

Š HomShAb S.coinv.C �

/; res.I �

//

Š RHomShAb S.Lcoinv.Z/;W /:

4.2.6 In general, the coinvariants functor coinv.: : : / D ˝ZŒH� Z can be written interms of the tensor product in the sense of presheaves composed with a sheafification.Furthermore, C �.H/ is the sheafification of pC �.H/. Using the fact that the tensorproduct of presheaves commutes with sheafification

U i D coinv.C i /

Š .C i .H/˝pZŒH� Z/] (Definition 4.14)

D ..pC i .H//] ˝p.pZŒH�/]

. pZ/]/]

Š . pC i .H/˝ppZŒH�pZ/] (22)

Š . pZŒH �˝pZ pZ.H � �H„ ƒ‚ …i factors

/˝ppZŒH�pZ/] (Equation (20))

Š . pZ.H � �H„ ƒ‚ …i factors

//]

D . pDi /]

withpDi WD pZ.H � �H„ ƒ‚ …

i factors

/: (23)

In particular, we haveU i D Z.H i /: (24)

4.2.7 Let G be a group. Then we can form the standard reduced bar complex for thegroup homology with integer coefficients

G� ! Z.Gn/! Z.Gn�1/! ! Z! 0:

The abelian group Z.Gn/ (sitting in degree n) is freely generated by the underlying setof Gn, and we write the generators in the form Œg1j : : : jgn�. The differential is givenby

d DnXiD0.�1/idi W Z.Gn/! Z.Gn�1/;

where

di Œg1j : : : jgn� WD

8<:Œg2j : : : jgn�; i D 0;Œg1j : : : jgigiC1j : : : jgn�; 1 � i � n � 1;Œg1j : : : jgn�1�; i D n:

The cohomology of this complex is the group homology H�.GIZ/.


4.2.8 For A 2 S the complex pD�.A/ (see (23)) is exactly the standard complex (see4.2.7) for the group homology H�.H.A/;Z/ of the group H.A/. The cohomologysheaves H�.U �/ are thus the sheafifications of the cohomology presheaves

S 3 A 7! H�.H.A/;Z/ 2 Ab:

4.2.9 In this paragraph we collect some facts about the homology of abelian groups.An abelian group V is the same thing as a Z-module. We define the graded Z-algebraƒ�ZV as the quotient of the tensor algebra

TZV WDMn�0

V ˝Z ˝Z V„ ƒ‚ …n factors

by the graded ideal I � TZV generated by the elements x ˝ x, x 2 V .

4.2.10 Let G be an abelian group. We refer to [Bro82, V.6.4] for the following fact.

Fact 4.15. There exists a canonical map

m W ƒiZG ! Hi .GIZ/:It is an isomorphism for i D 0; 1; 2, and it becomes an isomorphism after tensoring with

Q for all i 0. If G is torsion-free, then it is an isomorphism m W ƒiZG�! Hi .GIZ/

for all i 0.

4.2.11 Let U � D U �.H/ (see Definition 4.14) for H 2 ShAb S. The cohomologysheaves L�coinv.Z// Š H�.U �/ are the sheafifications of the presheaves H�. pD�/.By the fact 4.15 we have H i . pD�/ Š ƒiZH for i D 0; 1; 2 for the presheaves S 3U 7! ƒiZH.U /. In particular, we have H 0.U �/ Š Z and H 1.U �/ Š H . If H isa torsion-free sheaf, then H i .U �/ Š .ƒiZH/

] for all i 0. Finally, for an arbitrarysheaf H 2 ShAb S we have

.ƒ�ZH/] ˝Z Q Š H�.U �

/˝Z Q:

4.2.12 Our application of this relies on the study of the two spectral sequences con-verging to

Ext�ShAb S.Lcoinv.Z/;Z/ Š Ext�ShZŒH�-modS.Z;Z/:

We choose an injective resolution Z! I � in ShAb S. Then

Ext�ShAb S.Lcoinv.Z/;Z/ Š H�.HomShAb S.U�

; I�

//:

The first spectral sequence denoted by .Fr ; dr/ is obtained by taking the cohomologyin the U �-direction first. Its second page is given by

Fp;q2 Š ExtpShAb S.L

qcoinv.Z/;Z/:


This page contains the object of our interest, namely by 4.2.11 the sheaves of groups

Fp;12 Š ExtpShAb S.H;Z/:

The other spectral sequence .Er ; dr/ is obtained by taking the cohomology in theI �-direction first. In view of (22) its first page is given by

Ep;q1 Š ExtqShAb S.Z.H

p/;Z/;

which can be evaluated easily in many cases.Let us note that

H�RHomShZŒH�-modS.Z;Z/ Š Ext�ShZŒH�-modS.Z;Z/

has the structure of a graded ring with multiplication given by the Yoneda product.

4.2.13 We now verify Assumption 2 of Lemma 4.9 for all compact groups.

Proposition 4.16. LetH 2 S be a compact group. Then we have ExtiShAb S.H;R/ Š 0for i D 1; 2.

As in Definition 4.14 let U � WD U �.H/. Let R ! I � be an injective resolution.Then we get a double complex HomShAb S.U

�; I �/.We first take the cohomology in the I �-, and then in the U �-direction. We get a

spectral sequence with first term

Ep;q1 Š ExtqShAb S.Z.FH

p/;R/:

It follows from Corollary 3.28, 2., that Ep;q1 Š 0 for q 1.We consider the complex

C�

.H;R/ W 0! Map.H;R/! ! Map.Hp�1;R/! Map.Hp;R/! of topological groups which calculates the continuous group cohomologyH�cont.H IR/of H with coefficients in R. Now observe that by the exponential law for A 2 S

Ext0ShAb S.Z.FHp/;R/.A/

Lemma 3.9Š HomS.Hp � A;R/ Š HomS.A;Map.Hp;R//:

Hence the complex .E�;01 ; d1/.A/ is isomorphic to the complex

HomS.A; C�

.H;R// D C �

.H;R/.A/:

SinceH is a compact group we haveH icont.H IR/ Š 0 for i 1. Of importance for

us is a particular continuous chain contraction hp W Map.Hp;R/ ! Map.Hp�1;R/,p 1, which is given by the following explicit formula. If c 2 Map.Hp;R/ is acocycle, then we can define hp.c/ WD b 2 Map.Hp�1;R/ by the formula

b.t1; : : : ; tp�1/ WD .�1/pZH

c.t1; : : : ; tp�1; t /dt;


where dt is the normalized Haar measure. Then we have db D c. The maps.hp/p>0 induce a chain contraction .hp� /p>0 of the complex C �.H;R/. Therefore

H iMap.A;C �.H;R// Š H iC �.H;R/.A/ for i 1, too.The spectral sequence thus degenerates from the second page on. We conclude that

H iHomShAb S.U�

; I�

/ Š 0; i 1:We now take the cohomology of the double complex Homi

ShAb S.U�; I �/ in the other

order, first in the U �-direction and then in the I �-direction. In order to calculate thecohomology ofU � in degree� 2we use the fact 4.15. We get again a spectral sequencewith second term (for q � 2, using 4.2.11)

Fp;q2 Š ExtpShAb S..ƒ

qZH/

];R/:

We know by Corollary 3.28, 2., that

Fp;02 Š ExtpShAb S.Z;R/ Š 0; p 1

(note that we can write Z D Z.¹�º/ for a one-point space). Furthermore note that

F0;12 Š HomShAb S.H;R/ Š Homtop-Ab.H;R/ Š 0

since there are no continuous homomorphisms H ! R. The second page of thespectral sequence thus has the following structure.

2 HomShAb S..ƒ2ZH/

];R/

1 0 Ext1ShAb S.H;R/ Ext2ShAb S.H;R/

0 R 0 0 0 0

0 1 2 3 4

Since the spectral sequence must converge to zero in positive degrees we see that

Ext1ShAb S.H;R/ Š 0:We claim that HomShAb S..ƒ

2ZH/

];R/ Š 0. Note that for A 2 S we have

HomShAb S..ƒ2ZH/

];R/.A/Š HomPrAb S.ƒ2ZH;R/.A/Š HomPrAb S=A.ƒ

2ZH jA;RjA/:

An element � 2 HomPrAb S=A.ƒ2ZH jA;RjA/ induces a family of a biadditive (antisym-

metric) maps�W W H.W / �H.W /! R.W /

for .W ! A/ 2 S=A which is compatible with restriction. Restriction to points givescontinuous biadditive maps H �H ! R. Since H is compact the only such map isthe constant map to zero. Therefore �W vanishes for all .W ! A/. This proves theclaim.

Again, since the spectral sequence .Fr ; dr/must converge to zero in higher degreeswe see that

F2;12 Š Ext2ShAb S.H;R/ Š 0:

This finishes the proof of the lemma.


4.3 Discrete groups

4.3.1 In this subsection we study admissibility of discrete abelian groups. First weshow the easy fact that a finitely generated discrete abelian group is admissible. Inthe second step we try to generalize this result using the representation of an arbitrarydiscrete abelian group as a colimit of its finitely generated subgroups. The functorExt�ShAb S.: : : ;T / does not commute with colimits because of the presence of higherR lim-terms in the spectral sequence (26), below.

And in fact, not every discrete abelian group is admissible.

Lemma 4.17. For n 2 N we have ExtiShAb S.Z=nZ;Z/ Š 0 for i 2.

Proof. We apply the functor Ext�ShAb S.: : : ;Z/ to the exact sequence

0! Z! Z! Z=nZ! 0

and get the long exact sequence

Exti�1ShAb S.Z;Z/! ExtiShAb S.Z=nZ;Z/! ExtiShAb S.Z;Z/! :

Since by Theorem 4.10 ExtiShAb S.Z;Z/ Š 0 for i 1, the assertion follows.

Theorem 4.18. A finitely generated abelian group is admissible.

Proof. The group Z=nZ is admissible, since Assumption 2 of Lemma 4.9 followsfrom Proposition 4.16, while Assumption 1 follows from Lemma 4.17. The group Zis admissible by Theorem 4.10 since we can write Z Š Z.¹�º/ for a point ¹�º2 S. Afinitely generated abelian group is a finite product of groups of the form Z and Z=nZfor various n 2 N. A finite product of admissible groups is admissible.

4.3.2 We now try to extend this result to general discrete abelian groups using col-imits. Let I be a filtered category (see [Tam94, 0.3.2] for definitions). The categoryShAb S is an abelian Grothendieck category (see [Tam94, Theorem I.3.2.1]). The cat-egory HomCat.I;ShAb S/ is again abelian and a Grothendieck category (see [Tam94,Proposition 0.1.4.3]). We have the adjoint pair

colim W HomCat.I;ShAb S/, ShAb S W C‹;where to F 2 ShAb S there is associated the constant functor CF 2 HomCat.I;ShAb S/with value F by the functor C‹. The functor C‹ also has a right-adjoint lim:

C‹ W ShAb S, HomCat.I;ShAb S/ W lim:This functor is left-exact and admits a right-derived functor R lim. Finally, for F 2HomCat.I;ShAb S/ we have the functor

IHomShAb S.F; : : : / W ShAb S! HomCat.Iop;ShAb S/


which fits into the adjoint pairZ I

˝ F W HomCat.Iop;ShAb S/,W ShAb S W IHomShAb S.F; : : : /; (25)

where forG 2 HomCat.Iop;ShAb S/ the symbol

R IG˝F 2 ShAb S denotes the coend

of the functor I op�I ! ShAb, .i; j / 7! G.i/˝F.j /; correspondinglyRI

denotes theend of the appropriate functor. Indeed we have for all A 2 HomCat.I

op;ShAb S/ andB 2 ShAb S a natural isomorphism

HomShAb S.

Z I

A˝ F;B/ ŠZI

I op�IHomShAb S.A˝ F;B/

ŠZI

I opHomShAb S.A;

IHomShAb S.F; B//

Š HomHomCat.I op;ShAb S/.A;IHomShAb S.F; B//

The functor IHomShAb S.F; : : : / is therefore also left-exact and admits a right-derived version.

4.3.3 We say that P 2 HomCat.I;ShAb S/ is I -free if there exists a collection offlat sheaves .U.l//l2I , U.l/ 2 ShAb S, such that P.j / D L

l!j U.l/, and P.i !j / W L

l!i U.l/ !Ll!j U.l/ maps the summand U.l/ at l ! i identically to the

summand U.l/ at l ! i ! j .

Lemma 4.19. If P 2 HomCat.I;ShAb S/ is I -free, and if J 2 ShAb S is injective, thenIHomShAb S.P; J / 2 HomCat.I

op;ShAb S/ is injective.

Proof. We consider an exact sequence

.A� W 0! A0 ! A1 ! A2 ! 0/

in HomCat.Iop;ShAb S/. Then by Equation (25) we have

HomHomCat.I op;ShAb S/.A�

; IHomShAb S.P; J // Š HomShAb S

� Z I

A� ˝ P; J

�:

Exactness in HomCat.Iop;ShAb S/ is defined object-wise [Tam94, Theorem 0.1.3.1].

Therefore 0 ! A0.i/ ! A1.i/ ! A2.i/ ! 0 is an exact complex of sheaves for alli 2 I . Since P.j / is flat for all j 2 I the complex A�.i/˝ P.j / is exact for all pairs.i; j / 2 I � I . The complex of sheaves

R IA� ˝ P is the complex of push-outs along

the exact sequence of diagrams

F.i!j /2Mor.I /A

�.j /˝ P.i/ id˝P.i!j /��

A�

.i!j /˝id

��

Fj2I A

�.j /˝ P.j /

Fi2I A

�.i/˝ P.i/.


We claim that thisR IA� ˝ P ŠL

i2I A�.i/˝ U.i/. Since U.i/ is flat for all i 2 I

this is a sum of exact sequences and hence exact. Since J is injective we conclude thatHomShAb S.

R IA�˝P; J / is exact. So HomHomCat.I op;ShAb S/.: : : ;

IHomShAb S.P; J // pre-serves exactness, hence IHomShAb S.P; J / is an injective object in HomCat.I

op;ShAb S/.In order to finish the proof it remains to show the claim. We expand the push-out

diagram by inserting the structure of P .

Li!j A

�.j /˝Ll!i U.l/

aWDid˝.i!j /��

bWDA�

.i!j /˝id

��

Ll!j A

�.j /˝ U.l/

��Ll!i A

�.i/˝ U.l/ d WDA�

.l!i/˝id��Ll2I A

�.l/˝ U.l/.

It suffices to show that the lower right corner has the universal property. The lowerhorizontal map has a split s W L

l2I A�.l/˝U.l/!L

l!i A�.i/˝U.l/ given by the

inclusion of summands induced by l 7! lid�! l . Two maps l ; r from the lower left

and upper right corner to some sheaf V which satisfy l ı b D r ı a must inducea unique map W L

l2I A�.l/˝ U.l/ ! V such that ı d D l and ı c D r .

We have now other choice than to define WD l ı s, and this map has the requiredproperties as can be checked by an easy diagram chaise.

Lemma 4.20. Every sheaf F 2 HomCat.I;ShAb S/ has an I -free resolution.

Proof. It suffices to show that there exists a surjection P ! F from an I -free sheaf.We start with the surjection Z.F /! F and note that Z.F /.i/ is flat for all i 2 I . Thenwe define the I -free sheafP byP.i/ WD ˚l!iZ.F /.l/, and the surjectionP ! Z.F /

by .l ! i/� W Z.F /.l/! Z.F /.i/ on the summand of P.i/ with index .l ! i/.

Lemma 4.21. We have a natural isomorphism in DC.ShAb S/

RHomShAb S.colim.F /;H/ Š R limRIHomShAb S.F;H/:

Proof. LetF 2 HomCat.I;ShAb S/. By Lemma 4.20 we can choose an I -free resolutionP � of F . Let H 2 ShAb S and H ! I � be an injective resolution. Then we have

RHomShAb S.colim F;H/ Š HomShAb S.colim F; I�

/:

Since the category ShAb S is a Grothendieck abelian category [Tam94, Theorem I.3.2.1]the functor colim is exact [Tam94, Theorem 0.3.2.1]. Therefore colimP �.F /! colimFis a quasi-isomorphism. It follows that

HomShAb S.colim F; I�

/ Š HomShAb S.colim P�

.F /; I�

/:


By Lemma 3.8 the restriction functor is exact and therefore commutes with colimits.We get

HomShAb S.colim P�

.F /; I�

/.A/ Š HomShAb S=A..colim P�

.F //jA; I �

jA/Š HomShAb S=A.colim.P �

.F /jA/; I �

jA/

Š lim IHomShAb S=A.P�

.F /jA; I �

jA/

Š lim .IHomShAb S.P�

.F /; I�

/.A//

Š .lim IHomShAb S.P�

.F /; I�

//.A/;

where in the last step we use that a limit of sheaves is defined object-wise. In otherwords

HomShAb S.colim F; I�

/ Š HomShAb S.colim P�

.F /; I�

/ Š lim IHomShAb S.P�

.F /; I�

/

Applying Lemma 4.19 we see that IHomShAb S.P�.F /; I �/ 2 HomCat.I;ShAb S/ is

injective, and

RHomShAb S.colim F;H/ Š lim IHomShAb S.P�

.F /; I�

/

Š R lim IHomShAb S.P�

.F /; I�

/

Š R limRIHomShAb S.F;H/:

4.3.4 Lemma 4.21 implies the existence of s spectral sequence with second page

Ep;q2 Š Rp limRqIHomShAb S.F;H/ (26)

which converges to GrR�HomShAb S.colim F;H/.

Lemma 4.22. Let I be a category and U 2 S. Then we have a commutative diagram

DC..ShAb S/Iop/

R�.U;::: /

��

R lim �� DC.ShAb S/

R�.U;::: /

��

DC.AbIop/

R lim �� DC.Ab/.

Proof. Since the limit of a diagram of sheaves is defined object-wise we have�.U; : : : /ılim D lim ı �.U; : : : /. We show that the two compositions R�.U; : : : / ıR lim ,R lim ıR�.U; : : : / are both isomorphic toR.�.U; : : : /ılim/ by showing that lim and�.U; : : : /preserve injectives.

The left-adjoint of the functor lim W .ShAb S/Iop ! ShAb S is the constant diagram

functor C::: W ShAb S! .ShAb S/Iop

which is exact. Therefore lim preserves injectives.The left-adjoint of the functor �.U; : : : / is the functor ˝Z Z.U /. Since Z.U /

is a sheaf of flat Z-modules (it is torsion-free) this functor is exact (object-wise andtherefore on diagrams). It follows that �.U; : : : / preserves injectives.


4.3.5 The general remarks on colimits above are true for all sites with finite products(because of the use of Lemma 3.8). In particular we can replace S by the site Slc-acyc oflocally acyclic locally compact spaces.

Lemma 4.23. Let F 2 .ShAb Slc-acyc/I op

. If for all acyclic U 2 Slc-acyc the canonicalmap limF.U /! R lim.F.U // is a quasi-isomorphism of complexes of abelian groups,then limF ! R limF is a quasi-isomorphism.

Proof. We choose an injective resolution F ! I � in .ShAb Slc-acyc/I op

. As in the proofof Lemma 4.22 for each U 2 Slc-acyc the complex I �.U / is injective. If we assumethat U is acyclic, the map F.U / ! I �.U / is a quasi-isomorphism in .Ab/I

op, and

R lim .F.U // Š lim.I �.U //. By assumption

.limF /.U / Š lim.F.U //! lim.I �

.U // Š .limI �

/.U / Š .R limF /.U /

is a quasi-isomorphism for all acyclic U 2 Slc-acyc. An arbitrary object A 2 Slc-acyc canbe covered by acyclic open subsets. Therefore limF ! limI � is an quasi-isomorphismlocally on A for each A 2 Slc-acyc. Hence limF ! R limF is an isomorphism inDC.ShAb Slc-acyc/ .

4.3.6 LetD be a discrete group. Then we let I be the category of all finitely generatedsubgroups of D. This category is filtered. Let F W I ! Ab be the “identity” functor.Then we have a natural isomorphism

D Š colim F:

By Theorem 4.18 we know that a finitely generated group G is admissible, i.e.

RqHomShAb S.G;T / Š 0; q D 1; 2:Note that by Lemma 3.3 we have colimF D D. Using the spectral sequence (26) weget for p D 1; 2 that

RpHomShAb S.D;T / Š Rp lim HomShAb S.F ;T /: (27)

Letg W Slc-acyc ! S be the inclusion. Since T andD belong to Slc-acyc using Lemma 3.38we also have

g�RpHomShAb S.D;T / Š RpHomShAb Slc-acyc.D;T /

Š Rp lim HomShAb Slc-acyc.F ;T /:

(28)

4.3.7 We now must study theRp lim-term. First we make the index category I slightlysmaller. Let xD � D ˝Z Q DW DQ be the image of the natural map i W D ! DQ,d 7! d ˝ 1. We observe that xD generates DQ. We choose a basis B � xD of theQ-vector space DQ and let ZB � xD be the Z-lattice generated by B . For a subgroupA � D let xA WD i.A/ � DQ. We consider the partially ordered (by inclusion) set

J WD ¹A � D j A finitely generated and Q xA \ ZB � xA and Œ xA W xA \ ZB� <1º:Here ŒG W H� denotes the index of a subgroup H in a group G. We still let A denotethe “identity” functor A W J ! Ab.


Lemma 4.24. We have D Š colim A.

Proof. It suffices to show that J � I is cofinal. Let A � D be a finitely generatedsubgroup. Choose a finite subset Nb � B such that the Q-vectorspace Q Nb � DQ

generated by Nb contains xA and choose representatives b in D of the elements of Nb.They generate a finitely generated group U � D such that xU D i.U / D Z Nb. We nowconsider the group G WD hU;Ai. This group is still finitely generated. Similarly, sinceQ xG D Q xU CQ xA D Q Nb we have

Q xG \ ZB D Q Nb \ ZB D Nb � xU � xG:Moreover, since Q xG D Q Nb D Q. xG \ ZB/, Œ xG W xG \ ZB� <1, i.e. G � J . On theother hand, by construction A � G.

On J we define the grading w W J ! N0 by

w.A/ WD jAtorsj C rk xAC Œ xA W xA \ ZB� for A 2 J :Lemma 4.25. The category J together with the grading w W J ! N0 is a directcategory in the sense of Definition 5.1.1 in [Hov99].

Proof. We must show that A � G implies w.A/ � w.G/, and that A ¤ G impliesw.A/ < w.G/. First of all we have Ators � Gtors and therefore jAtorsj � jGtorsj.Moreover we have xA � xG, hence rk xA � rk xG. Finally, we claim that the canonicalmap

xA=. Na \ ZB/! xG=. xG \ ZB/

is injective. In fact, we have xA \ . xG \ ZB/ D . xA \ xG/ \ ZB D xA \ ZB: It followsthat

j xA W xA \ ZBj � j xG W xG \ ZBj:Let now A � G and w.A/ D w.G/. We want to see that this implies A D G. First

note that the inclusion of finite groupsAtors ! Gtors is an isomorphism since both groupshave the same number of elements. It remains to see that xA! xG is an isomorphism.We have rk xA D rk xG. Therefore xA \ ZB D Q xA \ ZB D Q xG \ BZ D xG \ BZ.Now the equality Œ xA W xA \ ZB� D Œ xG W xG \ ZB� implies that xA D xG.

4.3.8 The category C.Ab/ has the projective model structure whose weak equiva-lences are the quasi-isomorphisms, and whose fibrations are level-wise surjections.By Lemma 4.25 the category J op with the grading w is an inverse category. OnC.Ab/J

opwe consider the inverse model structure whose weak equivalences are the

quasi-isomorphisms, and whose cofibrations are the object-wise ones. The fibrationsare characterized by a matching space condition which we will explain in the following.

For j 2 J let Jj � J be the category with non-identity maps all non-identity mapsin J with codomain j . Furthermore consider the functor

Mj W C.Ab/Jop restriction��! C.Ab/.Jj /

op lim��! C.Ab/


There is a natural morphism F.j /!MjF for all F 2 C.Ab/Jop

. The matching spacecondition asserts that a map F ! G in C.Ab/J

opis a fibration if and only if

F.j /! G.j / �MjG MjFis a fibration in C.Ab/, i.e. a level-wise surjection, for all j 2 J . In particular, F isfibrant if F.j /!MjF is a level-wise surjection for all j 2 J .

For X 2 C.Ab/Jop

the fibrant replacement X ! RX induces the morphism

limX ! limRX Š R limX:

Let now again A denote the identity functor A W J ! Ab for the discrete abeliangroup D.

Lemma 4.26. 1. For all U 2 S which are acyclic the diagram of abelian groupsHomShAb S.A;T /.U / 2 C.Ab/J

opis fibrant.

2.

lim HomShAb Slc-acyc.A;T /! R lim HomShAb Slc-acyc.A;T / 2 C.Ab/Jop

is a quasi-isomorphism.

Proof. The assertion 1. for locally compact acyclic U verifies the assumption of Lem-ma 4.23. Hence 2. follows from 1. We now concentrate on 1. We must show that

HomShAb S.A.j /;T /.U /!MjHomShAb S.A;T /.U / (29)

is surjective. We have

MjHomShAb S.A;T / Š lim HomShAb S.AjJj ;T / Š HomShAb S.colim AjJj ;T /:

The map (29) is induced by the map

colimAjJj ! F .j /: (30)

By Lemma 3.3 we have colim AjJj Š colim AjJj . The map colim AjJj ! A.j / isof course an injection.

We finish the argument by the following observation. Let H ! G be an injectivemap of finitely generated groups (we apply this withH WD colimAjJj andG WD F.j /).Then for U 2 S

HomShAb S.G;T /.U /! HomShAb S.H;T /.U /

is a surjection. In fact we have

HomShAb S.G;T /.U / Š Homtop-Ab.G;T /.U / (by Lemma 3.5)

Š yG.U /


(and a similar equation forH ). SinceH ! G is injective, its Pontrjagin dual yG ! yHis surjective. Because of the classification of discrete finitely generated abelian groups,yG and yH both are homeomorphic to finite unions of finite dimensional tori. BecauseU is acyclic, every map U ! yH lifts to yG. Therefore yG.U / ! yH.U / is surjective.

Corollary 4.27. We have Rp lim HomShAb Slc-acyc.A;T / Š 0 for all p 1.

Theorem 4.28. Every discrete abelian group is admissible on Slc-acyc.

Proof. This follows from (28) and Corollary 4.27.

4.3.9 We now present an example which shows that not every discrete group D isadmissible on S or Slc, using Corollary 4.41 to be established later.

Lemma 4.29. Let I be an infinite set, 1 6D n 2 N andD WD ˚IZ=nZ. ThenD is notadmissible on S or Slc.

Proof. We consider the sequence

0! Z=nZ! Tn! T ! 0 (31)

which has no global section, and the product of I copies of it

0!YI

Z=nZ!YI

T !YI

T ! 0: (32)

This sequence of compact abelian groups does not have local sections. In fact, an opensubset of

QI T always contains a subset of the form U DQ

I 0 T �V , where I 0 � I iscofinite and V �Q

InI 0 T is open. A section s W U !QI T would consist of sections

of the sequence (31) at the entries labeled with I 0.By Lemma 3.4 the sequence of sheaves associated to (32) is not exact. In view of

Corollary 4.41 below, the group 4

QI Z=nZ Š ˚IZ=nZ is not admissible on S or Slc.

4.4 Admissibility of the groups Rn and T n

Theorem 4.30. The group T is admissible.

Proof. Since T is compact,Assumption2:of Lemma 4.9 follows from Proposition 4.16.It remains to show the first assumption of Lemma 4.9. As we will see this follows fromthe following result.

Lemma 4.31. We have ExtiShAb S.R;Z/ D 0 for i D 1; 2; 3.

Let us assume this lemma for the moment. We apply Ext�ShAb S.: : : ;Z/ to the exactsequence

0! Z! R! T ! 0


and consider the following part of the resulting long exact sequence for i D 2; 3:

Exti�1ShAb S.Z;Z/! ExtiShAb S.T ;Z/! ExtiShAb S.R;Z/! ExtiShAb S.Z;Z/:

The outer terms vanish since RHomShAb S.Z; F / Š F for all F 2 ShAb S. Thereforeby Lemma 4.31

ExtiShAb S.T ;Z/ Š ExtiShAb S.R;Z/ Š 0;and this is Assumption 1. of 4.9.

It remains to prove Lemma 4.31.

4.4.1 Proof of Lemma 4.31.We choose an injective resolution Z ! I �. The sheafR gives rise to the homological complex U � introduced in 4.14. We get the doublecomplex HomShAb S.U

�; I �/ as in 4.2.12. As before we discuss the associated twospectral sequences which compute Ext�ShZŒR�-modS.Z;Z/.

4.4.2 We first take the cohomology in the I �-, and then in the U �-direction. In viewof (24), the first page of the resulting spectral sequence is given by

Ep;q1 Š ExtqShAb S.Z.F Rp/;Z/;

where F R denotes the underlying sheaf of sets of R. By Corollary 3.28, 1. we haveEp;q1 Š 0 for q 1.

We now consider the case q D 0. For A 2 S we haveEp;01 .A/ Š �.A�RpIZ/ Š�.AIZ/ for all p, i.e. Ep;01 Š Z. We can easily calculate the cohomology of thecomplex .E�;01 ; d1/ which is isomorphic to

0! Z0! Z

id! Z0! Z

id! Z! :We get

H i .E�;01 ; d1/ Š

´Z; i D 0;0; i 1:

The spectral sequence .Er ; dr/ thus degenerates at the second term, and

H iHomShAb S.U�

; I�

/ Š´

Z; i D 0;0; i 1: (33)

4.4.3 We now consider the second spectral sequence .F p;qr ; dr/ associated to the doublecomplex HomShAb S.U

�; I �/ which takes cohomology first in the U � and then in theI �-direction. Its second page is given by

Fp;q2 Š ExtpShAb S.H

qU�

;Z/:

Since R is torsion-free we can apply 4.2.11 and get


qZR/];Z/:


In particular we have ƒ0ZR Š Z and thus

Fp;02 Š ExtpShAb S.Z;Z/ Š 0

for p 1 (recall that ExtpShAb S.Z;H/ Š 0 for every H 2 ShAb S and p 1).Furthermore, since ƒ1ZR Š R, by Lemma 3.5 we have

F0;12 Š HomShAb S.R;Z/ Š Homtop-Ab.R;Z/ Š 0

since Homtop-Ab.R;Z/ Š 0.Here is a picture of the relevant part of the second page.

3 Hom ShAb S..ƒ3ZR/];Z/

2 Hom ShAb S..ƒ2ZR/];Z/ Ext1ShAb S..ƒ

2ZR/];Z/

1 0 Ext1ShAb S.R;Z/ Ext2ShAb S.R;Z/ Ext3ShAb S.R;Z/

0 Z 0 0 0 0 0

0 1 2 3 4 5

4.4.4 Let V be an abelian group. Recall the definition ofƒ�V from 4.2.9. If V has thestructure of a Q-vector space, then T �1Z V has the structure of a graded Q-vector space,and I � T �1Z V is a graded Q-vector subspace. Therefore ƒ�1Z V has the structure ofa graded Q-vector space, too.

4.4.5 We claim that

F0;i2 Š HomShAb S..ƒ

iZR/];Z/ Š 0

for i 1. Note that

HomShAb S..ƒiZR/];Z/ Š HomPrAb S.ƒ

iZR;Z/:

Let A 2 S. An element

� 2 HomPrAb S.ƒiZR;Z/.A/ Š HomPrAb S=A.ƒ

iZRjA;ZjA/

induces a homomorphism of groups �W W ƒiZR.W /! Z.W / for every .W ! A/ 2S=A. SinceƒiZR.W / is a Q-vector space and Z.W / does not contain divisible elementswe see that �W D 0. This proves the claim.

The claim implies that F 2;12 Š Ext2ShAb S.R;Z/ survives to the limit of the spectralsequence. Because of (33) it must vanish. This proves Lemma 4.31 in the case i D 2.

The term F1;12 Š Ext1ShAb S.R;Z/ also survives to the limit and therefore also

vanishes because of (33). This proves Lemma 4.31 in the case i D 1.


4.4.6 Finally, since F 0;33 Š 0, we see that d2 W F 1;22 ! F3;12 must be an isomorphism,

i.e.d2 W Ext1ShAb S..ƒ

2ZR/];Z/

�! Ext3ShAb S.R;Z/:

We will finish the proof of Lemma 4.31 for i D 3 by showing that d2 D 0.

4.4.7 We consider the natural action of Zmult on R and hence on R. This turns R intoa sheaf of Zmult-modules of weight 1 (see 3.5.1). It follows that HomShAb S.R; I

�/ is acomplex of sheaves of Zmult-modules of weight 1. Finally we see that ExtiShAb S.R;Z/are sheaves of Zmult-modules of weight 1 for i 0.

Now observe (see 3.5.5) thatƒ2ZR is a presheaf of Zmult-modules of weight2. Hence.ƒ2ZR/] and thus Ext1ShAb S..ƒ

2ZR/];Z/ are sheaves of Zmult-modules of weight 2.

Since R ! U �.R/ DW U � is a functor we get an action of Zmult on U � and henceon the double complex HomShAb S.U

�; I �/. This implies that the differentials of theassociated spectral sequences commute with the Zmult-actions (see 4.6.11 for moredetails). This in particular applies to d2. The equality d2 D 0 now directly followsfrom the following lemma.

Lemma 4.32. Let V;W 2 ShAb S be sheaves of Zmult-modules of weights k 6D l .Assume thatW has the structure of a sheaf of Q-vector spaces. Ifd 2 HomShAb S.V;W /

is Zmult-equivariant, then d D 0.

Proof. Let ˛ W Zmult ! EndShAb S.V / and ˇ W Zmult ! EndShAb S.W / denote the ac-tions. Then we have d ı ˛.q/ � ˇ.q/ ı d D 0 for all q 2 Zmult. We consider q WD 2.Then .2k � 2l/ ı d D 0. SinceW is a sheaf of Q-vector spaces and .2k � 2l/ 6D 0 thisimplies that d D 0.

Theorem 4.33. The group R is admissible.

Proof. The outer terms of the exact sequence

0! Z! R! T ! 0

are admissible by Theorem 4.30 and Theorem 4.18. We now apply Lemma 4.5, stabilityof admissibility under extensions.

4.5 Profinite groups

Definition 4.34. A topological group G is called profinite if there exists a small left-filtered8 poset9 I and a system F 2 AbI such that

1. for all i 2 I the group F.i/ is finite,

2. for all i � j the morphism F.i/! F.j / is a surjection,

3. there exists an isomorphism G Š lim i2IF.i/ as topological groups.

8i.e. for every pair i; k 2 I there exists j 2 I with j � i and j � k9A poset is considered here as a category.


For the last statement we consider finite abelian groups as topological groups withthe discrete topology. Note that the homomorphisms G ! F.i/ are surjective for alli 2 I . We call the system F 2 AbI an inverse system.

Lemma 4.35 ([HR63]). The following assertions on a topological abelian group Gare equivalent:

1. G is compact and totally disconnected.

2. Every neighbourhood U � G of the identity contains a compact subgroup Ksuch that G=K is a finite abelian group.

3. G is profinite.

Lemma 4.36. LetG be a profinite abelian group and n 2 Z. We define the groupsK;Qas the kernel and cokernel of the multiplication map by n, i.e. by the exact sequence

0! K ! Gn! G ! Q! 0: (34)

Then K and Q are again profinite.

Proof. We writeG WD limj2JGj for an inverse system .Gj /j2J of finite abelian groups.We define the system of finite subgroups .Kj /j2J by the sequences

0! Kj ! Gjn! Gj :

Since taking kernels commutes with limits the natural projections K ! Kj , j 2 Jrepresent K as the limit K Š limj2JKj .

Since cokernels do not commute with limits we will use a different argument forQ.Since G is compact and the multiplication by n is continuous, nG � G is a closedsubgroup. Therefore the group theoretic quotient Q is a topological group in thequotient topology.

A quotient of a profinite group is again profinite [HM98], Exercise E.1.13. Here isa solution of this exercise, using the following general structural result about compactabelian groups.

Lemma 4.37 ([HR63]). If H is a compact abelian group, then for every open neigh-bourhood 1 2 U � H there exists a compact subgroup C � U such that H=C ŠTa � F for some a 2 N0 and a finite group F .

Since G is compact, its quotient Q is compact, too. This lemma in particularimplies that Q is the limit of the system of these quotients Q=C . In our case, since Gis profinite, it can not have T a as a quotient, i.e given a surjection

G ! Q! Ta � Fwe conclude that a D 0. Hence we can writeQ as a limit of an inverse system of finitequotients. This implies that Q is profinite.


4.5.1 Let0! K ! G ! H ! 0

be an exact sequence of profinite groups, where K ! G is the inclusion of a closedsubgroup.

Lemma 4.38. The sequence of sheaves

0! K ! G ! H ! 0

is exact.

Proof. By [Ser02, Proposition 1] every surjection between profinite groups has a sec-tion. Hence we can apply Lemma 3.4.

In this result one can in fact drop the assumptions thatK andG are profinite. In ourbasic example the group K is the connected component G0 � G of the identity of G.

Lemma 4.39. Let0! K ! G ! H ! 0

be an exact sequence of compact abelian groups such that H is profinite. Then thesequence of sheaves

0! K ! G ! H ! 0

is exact.

Proof. We can apply [HM98, Theorem 10.35] which says that the projection G ! H

has a global section. Thus the sequence of sheaves is exact by 3.4 (even as sequenceof presheaves).

4.5.2 In order to show that certain discrete groups are not admissible we use thefollowing lemma.

Lemma 4.40. Let0! K ! G ! H ! 0

be an exact sequence of compact abelian groups. If the discrete abelian group yK isadmissible, then sequence of sheaves

0! K ! G ! H ! 0

is exact.

Proof. If U is a compact group, then its Pontrjagin dual yU WD Homtop-Ab.U;T / is adiscrete group. Pontrjagin duality preserves exact sequences. Therefore we have theexact sequence of discrete groups

0! yH ! yG ! yK ! 0:


The surjective map of discrete sets yG ! yK has of course a section. Therefore byLemma 3.4 the sequence of sheaves of abelian groups

0! yH ! yG ! yK ! 0

is exact. We apply HomShAb S.: : : ;T / and get the exact sequence

0! HomShAb S.yK;T /! HomShAb S.

yG;T /! HomShAb S.

yH;T /! Ext1ShAb S.yK;T /! :

By Lemma 3.5 we have HomShAb S.yG;T / Š G etc. Therefore this sequence translates

to0! K ! G ! H ! Ext1ShAb S.

yK;T /! : (35)

By our assumption Ext1ShAb S.yK;T / D 0 so that G ! H is surjective.

4.5.3 The same argument does apply for the site Slc. Evaluating the surjectionG ! H

on H we conclude the following fact.

Corollary 4.41. If K � G is a closed subgroup of a compact abelian group suchthat yK is admissible or admissible over Slc, then the projection G ! G=K has localsections.

4.5.4 If G is an abelian group, then let Gtors � G denote the subgroup of torsionelements. We call G a torsion group, if Gtors D G. If Gtors Š 0, then we say that G istorsion-free. If G is a torsion group and H is torsion-free, then HomAb.H;G/ Š 0.

A presheaf F 2 PrS S is called a presheaf of torsion groups if F.A/ is a torsiongroup for every A 2 S. The notion of a torsion sheaf is more complicated.

Definition 4.42. A sheaf F 2 ShAb S is called a torsion sheaf if for each A 2 S andf 2 F.A/ there exists an open covering .Ui /i2I of A such that fjUi 2 F.Ui /tors.

The following lemma provides equivalent characterizations of torsion sheaves.

Lemma 4.43 ([Tam94], (9.1)). Consider a sheafF 2 ShAb S. The following assertionsare equivalent.

1. F is a torsion sheaf.

2. F is the sheafification of a presheaf of torsion groups.

3. The canonical morphism colimn2N.nF /!F is an isomorphism, where .nF /n2N

is the direct system nF WD ker.FnŠ! F /.

Note that a subsheaf or a quotient of a torsion sheaf is again a torsion sheaf.

Lemma 4.44. If H is a discrete torsion group, then H is a torsion sheaf.


Proof. We write H D colimn2N.nH/, where nH WD ker.HnŠ! H/. By Lemma 3.3

we have H D colimn2NnH . Now colimn2NnH is the sheafification of the presheafpcolimn2NnH of torsion groups and therefore a torsion sheaf.

In generalH is not a presheaf of torsion groups. Consider e.g.H WD ˚n2NZ=nZ.Then the element id 2 H.H/ is not torsion.

Definition 4.45. A sheaf F 2 ShAb S is called torsion-free if the group F.A/ is torsion-free for all A 2 S.

A sheaf F is torsion-free if and only if for all n 2 N the map Fn! F is injective.

It suffices to test this for all primes.

Lemma 4.46. If F 2 ShAb S is a torsion sheaf and E 2 ShAb S is torsion-free, thenHomShAb S.F;E/ Š 0.

Proof. We have

HomShAb S.F;E/ Š HomShAb S.colimn2N.nF /;E/ Š limn2NHomShAb S.nF;E/:

On the one hand, via the first entry multiplication by nŠ induces on HomShAb S.nF;E/

the trivial map. On the other hand it induces an injection since E is torsion-free.Therefore HomShAb S.nF;E/ Š 0 for all n 2 N. This implies the assertion.

4.5.5 If F 2 ShS and C 2 S, then we can form the sheaf RC .F / 2 ShS by theprescription RC .F /.A/ WD F.A � C/ for all A 2 S (see 3.3.18). We will show thatRC preserves torsion sheaves provided C is compact.

Lemma 4.47. If C 2 S is compact andH 2 ShAb S is a torsion sheaf, then RC .H/ isa torsion sheaf.

Proof. Given s 2 RC .H/.A/ D H.A � C/ and a 2 A we must find n 2 N and aneighbourhood U of a such that .ns/jU�C D 0. Since C is compact and A 2 S iscompactly generated by assumption on S, the compactly generated topology (this is thetopology we use here) on the productA�C coincides with the product topology ([Ste67,Theorem 4.3]). Since H is torsion there exists an open covering .Wi D Ai � Ci /i2Iof A � C and a family of non-zero integers .ni /i2I such that .nis/jWi D 0.

The set of subsets of C¹Ci j i 2 I; a 2 Aiº

forms an open covering of C . Using the compactness of C we choose a finite seti1; : : : ; ir 2 I with a 2 Aik such that ¹Cik j k D 1; : : : rº is still an open coveringof C . Then we define the open neighbourhood U of a 2 A by U WD Tr

kD1Aik . Setn WDQr

kD1 nik . Then we have .ns/U�C D 0.

Lemma 4.48. LetH be a discrete torsion group andG 2 S be a compactly generated 10

group. Then the sheaf HomShAb S.G;H/ is a torsion sheaf.

10i.e. generated by a compact subset


Proof. LetC � G be a compact generating set ofG. Precomposition with the inclusionC ! G gives an inclusion Homtop-Ab.G;H/! Map.C;H/. We have for A 2 S

HomShAb S.G;H/.A/ Š Homtop-Ab.G;H/.A/ (by Lemma 3.5)

Š HomS.A;Homtop-Ab.G;H//

� HomS.A;Map.C;H//

D HomS.A � C;H/Š H.A � C/Š RC .H/.A/:

By Lemma 4.44 the sheaf H is a torsion sheaf. It follows from Lemma 4.47 thatRC .H/ is a torsion sheaf. By the calculation above HomShAb S.G;H/ is a sub-sheafof the torsion sheaf RC .H/ and therefore itself a torsion sheaf.

Lemma 4.49. If G is a compact abelian group, then

Ext1ShAb S.G;Z/ Š HomShAb S.G;T / Š yG:Moreover, if G is profinite, then yG is a torsion sheaf.

Proof. We apply the functor Ext�ShAb S.G; : : : / to

0! Z! R! T ! 0

and get the following segment of a long exact sequence

! HomShAb S.G;R/! HomShAb S.G;T /

! Ext1ShAb S.G;Z/! Ext1ShAb S.G;R/! :SinceG is compact we have 0 D Homtop-Ab.G;R/ Š HomShAb S.G;R/ by Lemma 3.5.

Furthermore, by Proposition 4.16 we have Ext1ShAb S.G;R/ D 0. Therefore we get

Ext1ShAb S.G;Z/ Š HomShAb S.G;T / Š Homtop-Ab.G;T / Š yG;again by Lemma 3.5. If G is profinite, then (and only then) its dual yG is a discretetorsion group by [HM98, Corollary 8.5]. In this case, by Lemma 4.44 the sheaf yG is atorsion sheaf.

4.5.6 Let H be a discrete group. For A 2 S we consider the continuous group co-homology H i

cont.GIMap.A;H//, which is defined as the cohomology of the groupcohomology complex

0! Map.A;H/! HomS.G;Map.A;H//! ! HomS.Gi ;Map.A;H//!

with the differentials dual to the ones given by 4.2.7. The map

S 3 A 7! H icont.GIMap.A;H//

defines a presheaf whose sheafification we will denote by H i .G;H/.


Lemma 4.50. If G is profinite and H is a discrete group, then H i .G;H/ is a torsionsheaf for i 1.

Proof. We must show that for each section s 2 H icont.GIMap.A;H// and a 2 A there

exists a neighbourhood U of a and a number l 2 Z such that .ls/jU D 0. Thisadditional locality is important. Note that by the exponential law we have

C icont.G;Map.A;H// WD HomS.Gi ;Map.A;H// Š HomS.G

i � A;H/:Let s 2 H i

cont.GIMap.A;H// and a 2 A. Let s be represented by a cycle Os 2C icont.G;Map.A;H//. Note that Os W Gi�A! H is locally constant. The sets ¹Os�1.h/ jh 2 H º form an open covering of Gi � A.

Since G and therefore Gi are compact and A 2 S is compactly generated byassumption on S, the compactly generated topology (this is the topology we use here)on the product Gi � A coincides with the product topology ([Ste67, Theorem 4.3]).

SinceGi �¹aº� Gi �A is compact we can choose a finite set h1; : : : ; hr 2 H suchthat ¹Os�1.hi / j i D 1; : : : ; rº coversGi�¹aº. Now there exists an open neighbourhoodU � A of a such thatGi �U �Sr

sD1 Os�1.hi /. OnGi �U the function Os has at mosta finite number of values belonging to the set ¹h1; : : : ; hrº.

Since G is profinite there exists a finite quotient group G ! xG such that OsjGi�Uhas a factorization Ns W xGi � U ! H . Note that Ns is a cycle in C icont.

xG;Map.U;H//.Now we use the fact that the higher (i.e. in degree 1) cohomology of a finite

group with arbitrary coefficients is annihilated by the order of the group. Hence j xGjNs isthe boundary of some Nt 2 C i�1cont .

xG;Map.U;H//. Pre-composing Nt with the projectionGi �U ! xGi �U we get Ot 2 C icont.G;Map.U;H// whose boundary is Os. This showsthat .j xGjs/jU D 0.

4.5.7 Let G be a profinite group. We consider the complex U � WD U �.G/ as definedin 4.14. Let Z ! I � be an injective resolution. Then we get a double complexHomShAb S.U

�; I �/ as in 4.2.12.

Lemma 4.51. For i 1H iHomShAb S.U

�

; I�

/ is a torsion sheaf: (36)

Proof. We first take the cohomology in the I �-, and then in the U �-direction. The firstpage of the resulting spectral sequence is given by

Er;q1 Š ExtqShAb S.Z.G

r/;Z/:

Since the Gr are profinite topological spaces, by Lemma 3.29 we get Er;q1 Š 0 forq 1. We now consider the case q D 0. For A 2 S we have

Er;01 .A/ Š Z.A �Gr/ Š HomS.G

r � A;Z/ Š HomS.Gr ;Map.A;Z//

for all r 0. The differential of the complex .E�;01 .A/; d1/ is exactly the differential ofthe complexC icont.G;Map.A;Z// considered in 4.5.6. By Lemma 4.50 we conclude thatthe cohomology sheavesEi;02 are torsion sheaves. The spectral sequence degenerates atthe second term. We thus have shown thatH iHomShAb S.U

�; I �/ is a torsion sheaf.


4.5.8 For abelian groupsV;W letV �ZW WD TorZ1 .V;W / denote the Tor-product. IfV

andW in addition are Zmult-modules then V �W is a Zmult-module by the functorialityof the Tor-product.

Lemma 4.52. If V;W are Zmult-modules of weight k; l , then V ˝Z W and V �Z W

are of weight k C l .Proof. The assertion for the tensor product follows from (19). Let P � ! V 0 andQ� ! W 0 be projective resolutions of the underlying Z-modules V 0; W 0 of V;W .Then P �.k/! V and Q�.l/! W are Zmult-equivariant resolutions of V and W . Wehave by (19)

V �Z W Š H 1.P�

.k/˝Z Q�

.l// Š H 1.P� ˝Z Q

�

/.k C l/:4.5.9 The tautological action of Zmult on an abelian group G (we write G.1/ forthis Zmult-module) (see 3.5.1) induces an action of Zmult on the bar complex Z.G�/ bydiagonal action on the generators, and therefore on the group cohomologyH�.G.1/IZ/of G.

In the following lemma we calculate the cohomology of the group Z=pZ.1/ as aZmult-module.

Lemma 4.53. Let p 2 N be a prime. Then we have

H i .Z=pZ.1/IZ/ Š

8<:

Z.0/; i D 0;0; i odd,

Z=pZ.k/; i D 2k 2:Proof. The group cohomology of Z=pZ can be identified as a ring with the ring Z˚cZ=pZŒc�, where c has degree 2. Furthermore, for every group there is a canonicalmap yG ! H 2.GIZ/ which in the case G Š Z=pZ happens to be an isomorphism.This implies that H 2.Z=pZ.1/IZ/ has weight 1 as a Zmult-module. Since the cupproduct in group cohomology is natural it is Zmult-equivariant. Therefore, the powerck generates a module of weight k. Hence H 2k.Z=pZ.1/IZ/ has weight k.

4.5.10 For a Zmult-module V (with Zmult-action‰) letP vk

be the operator x 7! ‰vx�vkx. Note that this is a commuting family of operators. We let M v

23 be the monoidgenerated by P v2 ; P

v3 .

Definition 4.54. A Zmult-module V is called a weight 2-3-extension if for all x 2 Vand all v 2 Zmult there is P v 2M v

23 such that

P vx D 0: (37)

A sheaf of Zmult-modules is called a weight 2-3-extension, if every section locallysatisfies Equation (37) (with P v depending on the section and the neighborhood).

In the tables below we will mark weight 2-3-extensions with attribute weight 2-3.


Lemma 4.55. Every Zmult-module of weight 2 or of weight 3 is a weight 2-3-extension.The class of weight 2-3-extensions is closed under extensions.

Proof. This is a simple diagram chase, using the fact that for a Zmult-module W ofweight k, ‰vx D vkx for all k 2 Zmult and x 2 W .

Lemma 4.56. Let n 2 N and V be a torsion-free Z-module. For i 2 ¹0; 2; 3º thecohomology H i ..Z=pZ/n.1/IV / is a Z=pZ-module whose weight is given by thefollowing table.

i 0 2 3 4

weight 0 1 2 2-3

Moreover, H 1..Z=pZ/nIV / Š 0.

Proof. We first calculate H i ..Z=pZ/nIZ/ using the Künneth formula and inductionby n 1. The start is Lemma 4.53. Let us assume the assertion for products with lessthan n factors. The cases i D 0; 1; 2 are straightforward. We further get

H 3..Z=pZ/n.1/IZ/ Š H 2.Z=pZ.1/IZ/ �Z H2..Z=pZ/n�1.1/IZ/:

By Lemma 4.52 the �Z-product of two modules of weight 1 is of weight 2. Similarly,we have an exact sequence

0!4M

jD0H j .Z=pZn�1.1/IZ/˝H 4�j .Z=pZIZ/! H 4..Z=pZ/n.1/IZ/

!3M

jD2H j .Z=pZ.1/IZ/ �Z H

5�j ..Z=pZ/n�1.1/IZ/! 0:

By Lemma 4.55 and induction we conclude that H 4..Z=pZ/n.1/IZ/ is a weight 2-3-extension.

We can calculate the cohomology H i .GIV / of a group G in a trivial G-moduleby the standard complex C �.GIV / WD C.Map.G�; V //. If G is finite, then we haveC �.GIV / Š C �.GIZ/˝Z V . If V is torsion-free, it is a flat Z-module and thereforeH i .C �.GIZ/ ˝Z V / Š H i .C �.GIZ// ˝Z V . Applying this to G D .Z=pZ/n weget the assertion from the special case Z D V .

4.5.11 An abelian groupG is a Z-module. Let p 2 Z be a prime. If pg D 0 for everyg 2 G, then we say that G is a Z=pZ-module.

Definition 4.57. A sheaf F 2 ShAb S is a sheaf of Z=pZ-modules if F.A/ is a Z=pZ-module for all A 2 S.


4.5.12 Below we consider profinite abelian groups which are also Z=pZ modules.The following lemma describes their structure.

Lemma 4.58. Let p be a prime number. If G is compact and a Z=pZ-module, thenthere exists a set S such that G ŠQ

S Z=pZ. In particular, G is profinite.

Proof. The dual group yG of the compact groupG is discrete and also a Z=pZ-module,hence an Fp-vector space. Let S � yG be an Fp-basis. Then we can write yG Š˚SZ=pZ. Pontrjagin duality interchanges sums and products. We get

G Š yyG Š 3˚SZ=pZ ŠYS

1Z=pZ ŠYS

Z=pZ:

4.5.13 Assume that G is a compact group and a Z=pZ-module. Note that the con-struction G ! U � WD U �.G/ is functorial in G (see 4.6.11 for more details). Thetautological action of Zmult on G induces an action of Zmult on U �. We can improveLemma 4.51 as follows.

Lemma 4.59. Let Z! I � be an injective resolution. For i 2 ¹2; 3; 4º the sheaves

H iHomShAb S.U�

; I�

/

are sheaves of Z=pZ-modules and Zmult-modules whose weights are given by thefollowing table.

i 0 2 3 4

weight 0 1 2 2-3

Moreover, H 1HomShAb S.U�; I �/ Š 0.

Proof. We argue with the spectral sequence as in the proof of 4.51. Since Ep;q1 D 0

for q 1, it suffices to show that the sheaves Ei;02 are sheaves of Z=pZ-modulesand Zmult-modules of weight k, where k corresponds to i as in the table (and with theappropriate modification for i D 4).

We have for A 2 S

Ei;01 .A/ Š Z.A �Gi / Š HomS.G

i ;Map.A;Z// D C icont.GIMap.A;Z//;

where for a topological G-module V the complex C �

cont.GIV / denotes the continuousgroup cohomology complex. We now consider the presheaf zX i defined by

A 7! zX i .A/ WD H icont.G;Map.A;Z//:

Then by definition Ei;02 WD X i WD . zX i /] is the sheafification of zX i .The action of Zmult on G induces an action q 7! Œq� on the presheaf zX i , which

descends to an action of Zmult on the associated sheaf X i .We fix i 2 ¹0; 1; 2; 3º and let k be the associated weight as in the table. For i 2

we must show that for each section s 2 H icont.GIMap.A;Z// and a 2 A there exists a


neighbourhood U of a such that ..qk � Œq�/s/jU D 0 and .ps/jU D 0 for all q 2 Zmult.For i D 4, instead of ..qk � Œq�/s/jU D 0 we must find P v 2 M v

23 (depending on Uand s) (see 4.5.10 for notation) such that P vsjU D 0. For i D 1 we must show thatsjU D 0.

We perform the argument in detail for i D 2; 3. It is a refinement of the proof ofLemma 4.50. The cases i D 1 and i D 4 are very similar.

Let s be represented by a cycle Os 2 C icont.G;Map.A;Z//. Note that Os W Gi �A! Zis locally constant. The sets ¹Os�1.h/ j h 2 Zº form an open covering ofGi �A. SinceG and therefore Gi is compact and A 2 S is compactly generated by assumption onS, the compactly generated topology (this is the topology we use here) on the productGi � A coincides with the product topology ([Ste67, Theorem 4.3]). This allows thefollowing construction.

The set of subsets¹Os�1.h/ j h 2 Zº

forms an open covering of Gi � A. Using the compactness of the subset Gi � ¹aº�Gi � A we choose a finite set h1; : : : ; hr 2 Z such that Gi � ¹aº� Sr

iD1 Os�1.hi /.There exists a neighbourhood U � A of a such that Gi � U �Sr

iD1 Os�1.hi /.We now use that G is profinite by Lemma 4.58. Since Os.Gi � U/ is a finite set (a

subset of ¹h1; : : : ; hrº) there exists a finite quotient groupG ! xG such that OsjGi�U hasa factorization Ns W xGi �U ! Z. In our case xG Š .Z=pZ/r for a suitable r 2 N. Notethat Ns is a cycle inC icont.

xG;Map.U;Z//. Now by Lemma 4.56 we know that .qk� Œq�/Nsand p Ns are the boundaries of some Nt ; Nt1 2 C i�1cont .

xG;Map.U;Z//. Pre-composing Nt ; Nt1with the projection Gi � U ! xGi � U we get Ot ; Ot1 2 C icont.G;Map.U;Z// whoseboundaries are .qk � Œq�/Os and p Os, respectively. This shows that .qk � Œq�s/jU D 0

and .ps/jU D 0.

4.5.14 We consider a compact groupG and form the double complex HomShAb S.U�; I �/

defined in 4.5.7. Taking the cohomology first in the U � and then in the I �-direction weget a second spectral sequence .F p;qr ; dr/ with second page

Fp;q2 Š ExtpShAb S.H

qU�

;Z/:

In the following we calculate the term F1;22 and show the vanishing of several other

entries.

Lemma 4.60. The left lower corner of F p;q2 has the following form.

3 0

2 0 Ext1ShAb S..ƒ2ZG/

];Z/

1 0 Ext1ShAb S.G;Z/ Ext2ShAb S.G;Z/ Ext3ShAb S.G;Z/

0 Z 0 0 0 0 0

0 1 2 3 4 5


Proof. The proof is similar to the corresponding argument in the proof of 4.16. Using4.2.11 we get


qZG/

];Z/

for q D 0; 1; 2. In particular we have ƒ0ZG Š Z and thus

Fp;02 Š ExtpShAb S.Z;Z/ Š 0:

Furthermore, since ƒ1ZG Š G we have

F0;12 Š HomShAb S.G;Z/

Lemma 3.5Š Homtop-Ab.G;Z/ Š 0

since Homtop-Ab.G;Z/ Š 0 by compactness of G.We claim that

F0;22 Š HomShAb S..ƒ

2ZG/

];Z/ Š 0:Note that for A 2 S we have

HomShAb S..ƒ2ZG/

];Z/.A/ Š HomPrAb S.ƒ2ZG;Z/.A/ Š HomPrAb S=A.ƒ

2ZGjA;ZjA/:

An element � 2 HomPrAb S=A.ƒ2ZGjA;ZjA/ induces a family of biadditive (antisym-

metric) maps�W W G.W / �G.W /! Z.W /

for .W ! A/ 2 S=A which is compatible with restriction. Restriction to points givesbiadditive maps G � G ! Z. Since G is compact the only such map is the constantmap to zero. Therefore �W vanishes for all .W ! A/. This proves the claim. Thesame kind of argument shows that HomShAb S..ƒ

qZG/

];Z/.A/ D 0 for q 2.

Let us finally show that F 0;32 Š HomShAb S..H3U �/];Z/ Š 0. By 4.15 we have an

exact sequence (see (23) for the notation pD�)

0! K ! ƒ3ZG ! H 3. pD�

/! C ! 0;

where K and C are defined as the kernel and cokernel presheaf, and the middle mapbecomes an isomorphism after tensoring with Q. This means that 0 Š K ˝ Q and0 Š C ˝Q. Hence, theseK and C are presheaves of torsion groups. LetA 2 S. Thenan element s 2 HomShAb S..H

3U �/];Z/.A/ induces by precomposition an element Qs 2HomShAb S..ƒ

3ZG/

];Z/.A/ Š 0. Therefore s factors over Ns 2 HomShAb S.C];Z/.A/.

Since C is torsion we conclude that Ns D 0 by Lemma 4.46. The same argument showsthat HomShAb S..H

qU �/];Z/ Š 0 for q 3.

Lemma 4.61. Assume that p is an odd prime number, p ¤ 3. IfG is a compact groupand a Z=pZ-module, then ExtiShAb S.GIZ/ Š 0 for i D 2; 3.

If p D 3, then at least Ext2ShAb S.G;Z/ Š 0.


Proof. We consider the spectral sequence .Fr ; dr/ introduced in 4.5.14. It convergesto the graded sheaves associated to a certain filtrations of the cohomology sheaves ofthe total complex of the double complex HomShAb S.U

�; I �/ defined in 4.5.7. In degree2; 3; 4 these cohomology sheaves are sheaves of Z=pZ-modules carrying actions ofZmult with weights determined in Lemma 4.59.

The left lower corner of the second page of the spectral sequence was evaluated inLemma 4.60. Note that ExtiShAb S.G;Z/ is a sheaf of Z=pZ-modules with an action of

Zmult of weight 1 for all i 0. The term F2;12 Š Ext2ShAb S.G;Z/ survives to the limit

of the spectral sequence and is a submodule of a sheaf of Z=pZ-modules of weight 2.On the other hand it has weight 1.

A Z=pZ-module V with an action of Zmult which has weights 1 and 2 at the sametime must be trivial11. In fact, for every q 2 Zmult we get the identity q2 � q D.q � 1/q D 0 in EndZ.V /, and this implies that q � 1 .mod p/ for all q 62 pZ. Fromthis follows p D 2, and this case was excluded.

Similarly, the sheaf Ext1ShAb S..ƒ2ZG/

];Z/ is a sheaf of Z=pZ-modules of weight 2(see 3.5.5), and Ext3ShAb S.G;Z/ is a sheaf of Z=pZ-modules of weight 1. Since p is

odd, this implies that the differential d1;32 W Ext1ShAb S..ƒ2ZG/

];Z/ ! Ext3ShAb S.G;Z/

is trivial. Hence, Ext3ShAb S.G;Z/ survives to the limit. It is a subsheaf of a sheaf ofZ=pZ-modules which is a weight 2-3-extension. On the other hand it has weight 1.Substituting the weight 1-condition into equation (37), because then P v2 D v � v2 D.1 � v/v and P v3 D v � v3 D .1 � v/.1C v/v this implies that locally every sectionsatisfies .1 � v/n.1 C v/j vns D 0 for suitable n; j 2 N (depending on s and theneighborhood) and for all v 2 Zmult. If p > 3 we can choose v such that .1 � v/,.1C v/, v are simultaneously units, and in this case the equation implies that locallyevery section is zero.

We conclude that Ext3ShAb S.G;Z/ Š 0.

Lemma 4.62. Let G be a compact group which satisfies the two-three condition (seeDefinition 4.6). Assume further that

1. G is profinite, or

2. G is connected and locally topologically divisible (Definition 4.7).

Then the sheaves ExtiShAb S.G;Z/ are torsion-free for i D 2; 3.

Proof. We must show that for all primes p and i D 2; 3 the maps of sheaves

ExtiShAb S.G;Z/p! ExtiShAb S.G;Z/

are injective. The multiplication by p can be induced by the multiplication

p W G ! G:

11Note that in general a Z=pZ-module can very well have different weights. E.g a module of weight khas also weight pk.


We consider the exact sequence

0! K ! Gp! G ! C ! 0:

The groups K and C are groups of Z=pZ-modules and therefore profinite by 4.58.We claim that the sequence of sheaves

0! K ! Gp�! G ! C ! 0

is exact. We first show the claim in the case that G is profinite. Since C is profinite,by Lemma 4.39 and Lemma 3.4 we know that

0! G=K ! G ! C ! 0

is exact. Furthermore G=K is profinite so that the projection G ! G=K has localsections. This implies that G=K Š G=K, and hence the claim.

We now discuss the case of a connected locally topologically divisible G. In thiscase C D ¹1º. Since by assumption p W G ! G has local sections we can again useLemma 3.4 in order to conclude.

We decompose this sequence into two short exact sequences

0! K ! G ! Q! 0; 0! Q! G ! C ! 0:

Now we apply the functor Ext�ShAb S.: : : ;Z/ to the first sequence and study the associatedlong exact sequence

Exti�1ShAb S.G;Z/! Exti�1ShAb S.K;Z/! ExtiShAb S.Q;Z/! ExtiShAb S.G;Z/! :Note that Ext2ShAb S.K;Z/ Š 0 by Lemma 4.61 if p is odd, and by Lemma 4.17 if p D 2(the two-three condition ensures thatK in this case is an at most finite product of copiesof Z=2Z.). This implies that

Ext3ShAb S.Q;Z/! Ext3ShAb S.G;Z/

is injective.Next we show that

Ext2ShAb S.Q;Z/! Ext2ShAb S.G;Z/

is injective. For this it suffices to see that

Ext1ShAb S.G;Z/! Ext1ShAb S.K;Z/

is surjective. But this follows from the diagram

Ext1ShAb S.G;Z/ ��

Š��

Ext1ShAb S.K;Z/

Š��

yG surjective�� yK

;


where the vertical maps are the isomorphisms given by Lemma 4.49, and surjectivityof yG ! yK follows from the fact that Pontrjagin duality preserves exact sequences.

Next we apply Ext�ShAb S.: : : ;Z/ to the sequence

0! Q! G ! C ! 0

and get the long exact sequence

ExtiShAb S.C ;Z/! ExtiShAb S.G;Z/! ExtiShAb S.Q;Z/! :Again we use that ExtiShAb S.C ;Z/ Š 0 for i D 2; 3 by Lemma 4.61 for p > 3, andby Lemma 4.17 if p 2 ¹2; 3º (in this case C is an at most finite product of copies ofZ=pZ by the two-three condition) in order to conclude that

ExtiShAb S.G;Z/! ExtiShAb S.Q;Z/

is injective for i D 2; 3. Therefore the composition

ExtiShAb S.G;Z/! ExtiShAb S.Q;Z/! ExtiShAb S.G;Z/

is injective for i D 2; 3. This is what we wanted to show.

Lemma 4.63. Let G be a compact connected group which satisfies the two-three con-dition. Then the sheaves ExtiShAb Slc-acyc

.G;Z/ are torsion-free for i D 2; 3.

Proof. The only point in the proof of Lemma 4.62 where we have used the conditionof local topological divisibility was the exactness of the sequence

0! K ! Gp! G ! 0

of sheaves on S. We show that this sequence is exact with this condition if we considerthe sheaves on Slc-acyc (by restriction). Let us start with the dual sequence

0! yG p! yG ! yK ! 0

of discrete groups. It gives an exact sequence of sheaves

0! yG p! yG ! yK ! 0:

We apply HomShAb Slc-acyc.: : : ;T / and get, using Lemma 3.5, the long exact sequence

0! K ! Gp! G ! Ext1ShAb Slc-acyc

. yK;T /! :

By Theorem 4.28 we have Ext1ShAb Slc-acyc. yK;T / D 0. Now we can argue as in the proof

of 4.62.

Lemma 4.64. Let G be a profinite abelian group. Then the following assertions areequivalent.


1. ExtiShAb S.G;Z/ is torsion-free for i D 2; 32. ExtiShAb S.G;Z/ Š 0 for i D 2; 3.

Proof. It is clear that 2. implies 1. Therefore let us show that ExtiShAb S.G;Z/ Š 0 fori D 2; 3 under the assumption that we already know that it is torsion-free.

We consider the double complex HomShAb S.U�; I �/ introduced in 4.5.7. By Lem-

ma 4.51 we know that the cohomology sheavesH i .HomShAb S.U�; I �//of the associated

total complex are torsion sheaves.The spectral sequence .Fr ; dr/ considered in 4.5.14 calculates the associated graded

sheaves of a certain filtration of H i .HomShAb S.U�; I �//. The left lower corner of its

second page was already evaluated in Lemma 4.60.The term F

2;12 Š Ext2ShAb S.G;Z/ survives to the limit of the spectral sequence. On

the one hand by our assumption it is torsion-free. On the other hand by Lemma 4.51,case i D 2, it is a subsheaf of a torsion sheaf. It follows that

Ext2ShAb S.G;Z/ Š 0:This settles the implication 1:) 2: in case i D 2 of Lemma 4.64.

We now claim that Ext1ShAb S..ƒ2ZG/

];Z/ is a torsion sheaf. Let us assume the claim

and finish the proof of Lemma 4.64. Since F 3;12 Š Ext3ShAb S.G;Z/ is torsion-free by

assumption the differential d2 must be trivial by Lemma 4.46. Since also F 0;33 Š 0 thesheaf Ext3ShAb S.G;Z/ survives to the limit of the spectral sequence. By Lemma 4.51,case i D 3, it is a subsheaf of a torsion sheaf and therefore itself a torsion sheaf. Itfollows that Ext3ShAb S.G;Z/ is a torsion sheaf and a torsion-free sheaf at the same time,hence trivial. This is the assertion 1:) 2: of Lemma 4.64 for i D 3.

We now show the claim. We start with some general preparations. If F;H aretwo sheaves on some site, then one forms the presheaf S 3 A 7! .F ˝pZ H/.A/ WDF.A/ ˝Z H.A/ 2 Ab. The sheaf F ˝Z H is by definition the sheafification ofF ˝pZ H . We can write the definition of the presheaf ƒ2ZG in terms of the followingexact sequence of presheaves

0! K ! pZ.F .G//˛! G ˝pZ G ! ƒ2ZG ! 0;

whereK is by definition the kernel. ForW 2 S the map˛W W pZ.G.W //! G.W /˝Z

G.W / is defined on generators by ˛W .x/ D x ˝ x, x 2 G.W /. Sheafification is anexact functor and thus gives

0! . pZ.F .G//=K/] ! G ˝Z G ! .ƒ2ZG/] ! 0:

We now apply the functor HomShAb S.: : : ;Z/ and consider the following segment of theassociated long exact sequence

! HomShAb S.G ˝Z G;Z/! HomShAb S..pZ.F .G//=K/];Z/

! Ext1ShAb S..ƒ2ZG/

];Z/! Ext1ShAb S.G ˝Z G;Z/!(38)

The following two facts imply the claim:


1. Ext1ShAb S.G ˝Z G;Z/ is torsion.

2. HomShAb S..pZ.F .G//=K/];Z/ vanishes.

Let us start with 1. Let Z! I � be an injective resolution. We study

H 1HomShAb S.G ˝Z G; I�

/:

We have

HomShAb S.G ˝Z G; I�

/ Š HomShAb S.G;HomShAb S.G; I�

//:

Let K� D HomShAb S.G; I�/. Since HomShAb S.G;Z/ Š Homtop-Ab.G;Z/ Š 0 by the

compactness of G the map d0 W K0 ! K1 is injective. Let d1 W K1 ! K2 be thesecond differential. Now

H 1HomShAb S.G ˝Z G; I�

/ Š HomShAb S.G; ker.d1//=im.d0� /;

where d0� W HomShAb S.G;K0/ ! HomShAb S.G; ker.d1// is induced by d0. Since d0

is injective we haveim.d0� / D HomShAb S.G; im.d

0//:

Applying HomShAb S.G; : : : / to the exact sequence

0! K0 ! ker.d1/! ker.d1/=im.d0/! 0

and using ker.d1/=im.d0/ Š Ext1ShAb S.G;Z/ we get

0! HomShAb S.G;K0/

d0��! HomShAb S.G; ker.d1//!

! HomShAb S.G;Ext1ShAb S.G;Z//! Ext1ShAb S.G;K0/! :

In particular,

Ext1ShAb S.G ˝Z G;Z/ Š HomShAb S.G; ker.d1//=HomShAb S.G;K0/

� HomShAb S.G;Ext1ShAb S.G;Z//:

By Lemma 4.49 we know that Ext1ShAb S.G;Z/ Š yG. SinceG is compact and profinite,

the group yG is discrete and torsion. It follows by Lemma 4.48 that

HomShAb S.G;Ext1ShAb S.G;Z//

is a torsion sheaf. A subsheaf of a torsion sheaf is again a torsion sheaf. This finishesthe argument for the first fact.

We now show the fact 2.Note that

HomShAb S..pZ.F .G//=K/];Z/ Š HomShAb S.Z.F .G//=K

];Z/: (39)


Consider A 2 S and � 2 HomShAb S.Z.F .G//=K];Z/.A/. We must show that each

a 2 A has a neighbourhood U � A such that �jU D 0. Pre-composing � with theprojection Z.F .G//! Z.F .G//=K] we get an element

N� 2 HomShAb S.Z.F .G//;Z/.A/ Š HomSh S.F .G/;Z/.A/Lemma 3.9Š Z.A �G/:

We are going to show that N� D 0 after restriction to a suitable neighbourhood ofa 2 A using the fact that it annihilates K]. Let us start again on the left-hand side of(39). We have

HomShAb S..pZ.F .G//=K/];Z/.A/ Š HomPrAb S..

pZ.F .G//=K/;Z/.A/

Š HomPrAb S=A.pZ.F .GjA//=KjA;ZjA/:

For .A �G ! A/ 2 S=A the morphism � gives rise to a group homomorphism

y� W Z.F .G.A �G///=K.A �G/! Z.A �G/:The symbol F .G.A�G// denotes the underlying set of the groupG.A�G/. We have.prG W A�G ! G/ 2 G.A�G/ and by the explicit description of the element N� givenafter the proof of Lemma 3.9 we see N� D y� .¹ŒprG �º/, where ŒprG � 2 Z.F .G.A�G///denotes the generator corresponding to prG 2 G.A �G/, and ¹: : : º indicates that wetake the class modulo K.A �G/.

The homomorphism y� 2 HomAb.ZF .G.A � G//=K.A � G/;Z.A � G// is rep-resented by a homomorphism

Q� W ZF .G.A �G//! Z.A �G/;and we have N� D Q�.ŒprG �/. By definition of K we have the exact sequence

0! K.A�G/! Z.F .G.A�G///! G.A�G/˝ZG.A�G/! ƒ2ZG.A�G/! 0:

For x 2 G.A � G/ let Œx� 2 Z.F .G.A � G/// denote the corresponding generator.Since .nx/˝ .nx/ D n2.x ˝ x/ in G.A�G/˝Z G.A�G/ we have n2Œx�� Œnx� 2K.A � G/ for n 2 Z. It follows that Q� must satisfy the relation Q�.n2Œx�/ D Q�.Œnx�/for all n 2 Z and x 2 G.A � G/. Let us now apply this reasoning to x WD ŒprG �. For

every n 2 Z we have a map A � G idA�nDW n��! A � G, and by naturality the map Q�respects this action. Moreover, the element Œnx� 2 Z.F .G.A�G/// is obtained fromŒx� via this action, since

A �GprG��

n �� A �GprG��

Gn �� G

commutes. It follows that

n2 N� D Q�.n2ŒprG �/ D Q�.ŒnprG �/ D Q�.ZF . xG. n//.ŒprG �// D Z. n/. N�/ D n� N�;


where we write n� WD Z. n/. For k 2 Z we define the open subset Vk WD N��1.k/ �A�G. The family .Vk/k2K forms an open pairwise disjunct covering ofA�G. SinceGis compact the compactly generated topology ofA�G is the product topology ([Ste67,Theorem 4.3]). Since G is compact, we can choose a finite sequence k1; : : : ; kr 2 Zsuch that G � ¹aº�Sr

iD1 Vk . Furthermore, we can find a neighbourhood U � A of asuch that G � U �Sr

iD1 Vk .Note that N�jG�U has at most finitely many values. Therefore there exists a finite

quotient G ! F such that N�jU�G factors through f W U � F ! Z.The action of Z on G is compatible with the corresponding action of Z on F ,

and we have an action n� W HomS.U � F;Z/ ! HomS.U � F;Z/. The elementf 2 HomS.U � F;Z/ still satisfies n2f D n�f . We now take n WD jF j C 1. Thenn� D id so that .n2 � 1/f D 0, hence f D 0. This implies N�jU�G D 0.

This finishes the proof of the second fact.

4.5.15 The Lemma 4.62 verifies Assumption 1. in 4.64 for a large class of profinitegroups.

Corollary 4.65. If G is a profinite group which satisfies the two-three condition 4.6,then we have ExtiShAb S.G;Z/ Š 0 for i D 2; 3.

Theorem 4.66. A profinite abelian group which satisfies the two-three condition isadmissible.

4.6 Compact connected abelian groups

4.6.1 Let G be a compact abelian group. We shall use the following fact shown in[HM98, Corollary 8.5].

Fact 4.67. G is connected if and only if yG is torsion-free.

4.6.2 For a space X 2 S and F 2 PrAb S let LH�.X IF / denote the Cech cohomologyof X with coefficients in F . It is defined as follows. To each open covering U

one associates the Cech complex LC �.UIF /. The open coverings form a left-filteredcategory whose morphisms are refinements. The Cech complex depends functoriallyon the covering, i.e. if V ! U is a refinement, then we have a functorial chain mapLC �.UIF /! LC �.V IF /. We define

LC �

.X IF / WD colimULC �

.UIF /

and

LH�.X IF / WD colimUH�. LC �

.UIF // Š H�.colimULC �

.UIF // Š H�. LC �

.X IF //:


4.6.3 Fix a discrete group H . Let pH be the constant presheaf with values H . Notethat then the sheafification pH ] is isomorphic to H as defined in 3.2.5. Moreover

LC �

.X I pH/ Š LC �

.X;H/: (40)

In general Cech cohomology differs from sheaf cohomology. The relation betweenthese two is given by the Cech cohomology spectral sequence .Er ; dr/ (see [Tam94,3.4.4]) converging to H�.X IF /. Let H�.F / WD R�i.F / denote the right derivedfunctors of the inclusion i W ShAb S ! PrAb S. Then the second page of the spectralsequence is given by

Ep;q2 Š LHp.X IHq.F //:

By [Tam94, 3.4.7] the edge homomorphism

LHp.X IF /! Hp.X IF /is an isomorphism for p D 0; 1 and injective for p D 2.

4.6.4 We now observe that Cech cohomology transforms strict inverse limits of com-pact spaces into colimits of cohomology groups.

Lemma 4.68. Let H be a discrete abelian group. If .Xi /i2I is an inverse system ofcompact spaces in S such that X D lim i2IXi , then

LHp.X IH/ Š colim i2I LHp.Xi IH/:Proof. We first show that the system of open coverings of X contains a cofinal systemof coverings which are pulled back from the quotients pi W X ! Xi . Let U D .Ur/r2Rbe a covering by open subsets. For each r there exists a family Ir � I and subsetsUr;i � Xi , i 2 Ir such thatUr D [i2Irp�1i .Ur;i /. The set of open subsets ¹p�1i .Ur;i / jr 2 R; i 2 Irº is an open covering of X . Since X is compact we can choosea finite subcovering V WD ¹Ur1;i1 ; : : : Urk ;ik º which can naturally be viewed as arefinement of U. Since I is left filtered we can choose j 2 I such that j < idfor d D 1; : : : ; k. For j � i let pj i W Xj ! Xi be the structure map of the systemwhich are all surjective by the strictness assumption on the inverse system. ThereforeV 0 WD ¹p�1j;id .Urd ;id / j d D 1; : : : ; kº is an open covering of Xj , and V D p�1j .V 0/.

Now we observe that LC �.p�V 0I pH/ Š LC �.V 0I pH/. Therefore

LC �

.X IH/ Š LC �

.X I pH/ (by Equation (40))

Š colimULC �

.UI pH/D colim i2I colim coverings U ofXi

LC �

.UI pH/Š colim i2I LC �

.Xi I pH/Š colim i2I LC �

.Xi IH/ (by Equation (40))

since one can interchange colimits. Since filtered colimits are exact and thereforecommute with taking cohomology this implies the lemma.


4.6.5 Next we show that Cech cohomology has a Künneth formula.

Lemma 4.69. LetH be a discrete ring of finite cohomological dimension. Assume thatX; Y are compact. Then there exists a Künneth spectral sequence with second term

E2p;q WDMiCjDq

TorHp . LH i .X IH/; LH j .Y IH//

which converges to LHpCq.X � Y IH/.Proof. Since X is compact the topology on X � Y is the product topology [Ste67,Theorem 4.3]. Using again the compactness of X and Y we can find a cofinal systemof coverings of X � Y of the form p�U \ q�V for coverings U of X and V of Y ,where p W X �Y ! X and q W X �Y ! Y denote the projections, and the intersectionof coverings is the covering by the collection of all cross intersections. We have

LC �.p�U \ q�V I pH/ Š LC �.p�UI pH/˝H LC �.q�V I pH/:Since the tensor product over H commutes with colimits we get

LC �.X � Y I pH/ Š LC �.X I pH/˝H LC �.Y I pH/;and hence by (40)

LC �.X � Y IH/ Š LC �.X IH/˝H LC �.Y IH/:The Künneth spectral sequence is the spectral sequence associated to this double

complex of flat (as colimits of free) H -modules.

4.6.6 We now recall that sheaf cohomology also transforms strict inverse limits ofspaces into colimits, and that it has a Künneth spectral sequence. The following is aspecialization of [Brd97, II.14.6] to compact spaces.

Lemma 4.70. Let .Xi /i2I be an inverse system of compact spaces andX D lim i2IXi ,and H let be a discrete abelian group. Then

H�.X IH/ Š colim i2IH�.Xi IH/:For simplicity we formulate the Künneth formula for the sheaf Z only. The following

is a specialization of [Brd97, II.18.2] to compact spaces and the sheaf Z.

Lemma 4.71. For compact spaces X; Y we have a Künneth spectral sequence withsecond term

E2p;q DMiCjDq

TorZp .H

i .X IZ/;H j .Y IZ//

which converges to HpCq.X � Y IZ/.


Of course, this formulation is much too complicated since Z has cohomologicaldimension one. In fact, the spectral sequence decomposes into a collection of shortexact sequences.

Lemma 4.72. IfG is a compact connected abelian group, thenH�.GIZ/ Š LH�.GIZ/.

Proof. The Cech cohomology spectral sequence provides the map

LH�.GIZ/! H�.GIZ/:We now use that fact that G is the projective limit of groups isomorphic to T a, a 2 N.Since the compact abelian group G is connected, by Fact 4.67 we know that yG istorsion-free. Since yG is torsion-free it is the filtered colimit of its finitely generatedsubgroups yF . If yF � yG is finitely generated, then yF Š Za for some a 2 N, therefore

F WD yyF Š T a. Pontrjagin duality transforms the filtered colimit yG Š colim yF yF intoa strict limit G Š lim yFF .

Since T a is a manifold it admits a cofinal system of good open coverings where allmultiple intersections are contractible. Therefore we get the isomorphism

LH�.T aIZ/ �! H�.T aIZ/:Since both cohomology theories commute with limits of compact spaces we get

LH�.GIZ/ �! H�.GIZ/:

4.6.7 We now use the calculation of cohomology of the underlying space of a connectedcompact abelian group G. By [HM98, Theorem 8.83] we have

LH�.G;Z/ Š ƒ�Z yG (41)

as a graded Hopf algebra. This result uses the two properties 4.68 and 4.69 of Cechcohomology in an essential way.

By Lemma 4.72 we also have

H�.GIZ/ Š ƒ�Z yG:

Note that ƒ�Z yG is torsion free. In fact, ƒ�Z yG Š colim yFƒ�ZyF , where the colimit

is taken over all finitely generated subgroups yF of yG. We therefore get a colimit ofinjections of torsion-free abelian groups, which is itself torsion-free.

4.6.8 Recall the notation related to Zmult-actions introduced in 3.5.

Lemma 4.73. If V;W 2 Ab with W torsion-free and k; l 2 Z, k 6D l , then

HomZmult-mod.V .k/;W.l// D 0:


Proof. Let � 2 HomAb.V;W / and v 2 V . Then for all m 2 Zmult we have

mk�.v/ D ‰m.�.v// D �.‰m.v// D �.mlv/ D ml�.v/;i.e. each w 2 im.�/ � W.l/ satisfies .mk �ml/w D 0 for all m 2 Zmult. This set ofequations implies that w D 0, since W is torsion-free.

Lemma 4.74. If V;W 2 ShAb S with W torsion-free and k; l 2 Z, k 6D l , then

HomShZmult -modS.V .k/;W.l// D 0:We leave the easy proof of this sheaf version of Lemma 4.73 to the interested reader.

Note that a subquotient of a sheaf of Zmult-modules of weight k also has weight k.

4.6.9 By Slc � S we denote the sub-site of locally compact objects. The restriction tolocally compact spaces becomes necessary because of the use of the Künneth (or basechange) formula below.

The first page of the spectral sequence .Er ; dr/ introduced in 4.2.12 is given by

Eq;p1 D ExtpShAb S.Z.G

q/;Z/:

This sheaf is the sheafification of the presheaf

S 3 A 7! Hp.A �Gq;Z/ 2 Ab:

In order to calculate this cohomology we use the Künneth formula [Brd97, II.18.2] andthat H�.G;Z/ is torsion-free (4.67). We get for A 2 Slc that

H�.A �Gq;Z/ Š H�.A;Z/˝Z .ƒ� yG/˝Zq

for locally compact A. The sheafification of Slc 3 A! H i .A;Z/ vanishes for i 1,and gives Z for i D 0. Since sheafification commutes with the tensor product with afixed group we get

.Eq;�1 /jSlc Š .ƒ� yG/˝Zq : (42)

4.6.10 We now consider the tautological action of the multiplicative semigroup Zmult

on G of weight 1. As before we write G.1/ for the group G with this action. Observethat this action is continuous. Applying the duality functor we get an action of Zmult

on the dual group yG which is also of weight 1. Therefore

bG.1/ D yG.1/:The calculation of the cohomology (41) of the topological space G with Z-coeffi-

cients is functorial in G. We conclude thatH�.G.1/;Z/ Š ƒ�Z. yG.1// is a decompos-able Zmult-module. By 3.5.5 the group

Hp.Gq.1/;Z/ Š Œ.ƒ� yG.1//˝Zq�p

is of weight p.


4.6.11 We now observe that the spectral sequence .Er ; dr/ is functorial in G. To thisend we use the fact that G 7! U �.G/ is a covariant functor from groups G 2 S tocomplexes of sheaves on S. If G0 ! G1 is a homomorphism of topological groups inS, then we get an induced mapU q.G0/! U q.G1/, namely the map Z.Gq0/! Z.Gq1/.Under the identification made in 4.6.9 the induced map

ExtpShAb Slc.Z.Gq1/;Z/! ExtpShAb Slc

.Z.Gq0/;Z/

goes to the mapHp.G

q1 IZ/! Hp.G

q0 IZ/

induced by the pull-back

Hp.Gq1 IZ/! Hp.G

q0 IZ/

associated to the map of spaces Gq0 ! Gq1 .

4.6.12 The discussion in 4.6.11 and 4.6.10 shows that the Zmult-module structureG.1/induces one on the spectral sequence .Er ; dr/, and we see that Eq;p1 has weight p(which is the number of factors yG.1/ contributing to this term).

We introduce the notation H k WD H k.HomShAb S.U�; I �//jSlc . Note that H k has a

filtration0 D F �1H k � F 0H k � � F kH k D H k

which is preserved by the action of Zmult. The spectral sequence .Er ; dr/ converges tothe associated graded sheaf Gr.H k/.

Note that Grp.H k/ is a subquotient of Ek�p;p2 . Since a subquotient of a sheaf ofweight p also has weight p (see the our remark after Lemma 4.74) we get the followingconclusion.

Corollary 4.75. Grp.H k/ has weight p.

4.6.13 We now study some aspects of the second page Eq;p2 of the spectral sequencein order to show the following lemma.

Lemma 4.76. 1. For k 1 we have F 0H k Š 0.2. For k 2 we have F 1H k Š 0.

Proof. We start with 1. We see that E�;02 is the cohomology of the complex E�;01 ŠHomShAb S.U

�;Z/. Explicitly, using in the last step the fact that G and therefore Gq

are connected,

Eq;01 Š HomShAb S.U

q;Z/(24)Š HomShAb S.Z.G

q/;Z/(17)Š RGq .Z/

Š Map.Gq;Z/ Š Z (by Lemma 3.25):


An inspection of the formulas 4.2.7 for the differential of the complex U � shows thatd1 W Eq;01 ! E

qC1;01 vanishes for even q, and is the identity for odd q. In other words

this complex is isomorphic to

0! Z0! Z

id! Z0! Z

id! Z0! :

This implies that Eq;02 D 0 for q 1 and therefore the assertion of the lemma.We now turn to 2. We calculate

H 1.GqIZ/ Š Œ.ƒ� yG/˝Zq�1 D yG ˚ ˚ yG„ ƒ‚ …q summands

:

By (42) we getEq;11 Š yGq :

We see that .E�;11 ; d1/ is the sheafification of the complex of discrete groups

yG� W 0! yG ! yG2 ! yG3 ! ; (43)

where the differential is the dual of the differential of the complex

0 G G2 G3 induced by the maps given in 4.2.7. Let us describe the differential more explicitly. If� 2 yG, and � W G �G ! G is the multiplication map, then we have �� D .�; �/ 22G �G Š yG � yG. We can write @ W yGq ! yGqC1 as

@ DqC1XiD0

.�1/i@i :

Using the formulas of 4.2.7 we get

@i .�1; : : : ; �q/ D .�1; : : : ; �i ; �i ; : : : ; �q/for i D 1; : : : ; q. Furthermore

@0.�1; : : : ; �q/ D .0; �1; : : : ; �q/and

@qC1.�1; : : : ; �q/ D .�1; : : : ; �q; 0/:Note that 0 D @ W yG0 ! yG1. We see that we can write the higher ( 1) degree part ofthe complex (43) in the form

K� ˝Z yG;

whereKq D Zq for q 1 and @ W Zq ! ZqC1 is given by the same formulas as above.One now shows12 that

H q.K�

/ D´

Z; q D 1;0; q 2:

12We leave this as an exercise in combinatorics to the interested reader.


Since yG is torsion-free and hence a flat Z-module we have

H q. yG�

/ Š H q.K� ˝Z yG/ Š H q.K

�

/˝Z yG:Therefore H q. yG�/ D 0 for q 2. This implies the assertion of the lemma in thecase 2.

Lemma 4.77. Let F 0 � F 1 � � F k�1 � F k D F be a filtered sheaf of Zmult-modules such that Grl.F / has weight l , and such that F 0 D F 1 D 0. If V � F is atorsion-free sheaf of weight 1, then V D 0.

Proof. Assume that V 6D 0. We show by induction (downwards) that F l \ V 6D 0 forall l 1. The case l D 1 gives the contradiction. Assume that l > 1. We consider theexact sequence

0! F l�1 \ V ! F l \ V ! Grl.F /:

First of all, by induction assumption, the sheaf V \ F l is non-trivial, and a as asubsheaf of V it is torsion-free of weight 1. Since Grl.F / has weight l 6D 1, the mapV \ Fl ! Grl.F / can not be injective. Otherwise its image would be a torsion-freesheaf of two different weights l and 1, and this is impossible by Lemma 4.74. HenceF l�1 \ V 6D 0.

4.6.14 We now show that Lemma 4.64 extends to connected compact groups.

Lemma 4.78. Let G be a compact connected abelian group. Then the followingassertions are equivalent.

1. ExtiShAb Slc.G;Z/ is torsion-free for i D 2; 3.

2. ExtiShAb Slc.G;Z/ Š 0 for i D 2; 3.

Proof. The non-trivial direction is that 1. implies 2. Therefore let us assume thatExtiShAb Slc

.G;Z/ is torsion-free for i D 2; 3. We now look at the spectral sequence.Fr ; dr/. The left lower corner of its second page was calculated in 4.60. We seethat F 1;22 D Ext1ShAb Slc

..ƒ2ZG/];Z/ has weight 2, while Ext3ShAb Slc

.G;Z/ has weight 1.Since Ext3ShAb Slc

.G;Z/ is torsion-free by assumption, it follows from Lemma 4.74 that

the differential d1;22 W Ext1ShAb Slc..ƒ2ZG/

];Z/! Ext3ShAb Slc.G;Z/ is trivial.

We conclude that Ext2ShAb Slc.G;Z/ and Ext3ShAb Slc

.G;Z/ survive to the limit of thespectral sequence. We see that ExtiShAb Slc

.G;Z/ are torsion-free sub-sheaves of weight1 ofH iC1 for i D 2; 3. Using the structure ofH iC1 given in 4.75 in conjunction withLemmas 4.76 and 4.77 we get ExtiShAb Slc

.G;Z/ D 0 for i D 2; 3.

4.6.15 The combination of Lemma 4.62 (resp. Lemma 4.62) and Lemma 4.78 givesthe following result.

Theorem 4.79. 1. A compact connected abelian groupG which satisfies the two-threecondition is admissible on the site Slc-acyc.

2. If G is in addition locally topologically divisible, then it is admissible on Slc.


5 Duality of locally compact group stacks

5.1 Pontrjagin Duality

5.1.1 In this subsection we extend Pontrjagin duality for locally compact groups toabelian group stacks whose sheaves of objects and automorphisms are represented bylocally compact groups. In algebraic geometry a parallel theory has been consideredin [DP].

The site S denotes the site of compactly generated spaces as in 3.1.2 or one of itssub-sites Slc, Slc-acyc. The reason for considering these sub-sites lies in the fact thatcertain topological groups are only admissible on these sub-sites (see the Definitions4.1, 4.2 and Theorem 4.8).

5.1.2 Let F 2 ShAbS.

Definition 5.1. We define the dual sheaf of F by

D.F / WD HomShAbS.F;T /:

Definition 5.2. We call F dualizable, if the canonical evaluation morphism

c W F ! D.D.F // (44)

is an isomorphism of sheaves.

Lemma 5.3. If G is a locally compact group which together with its Pontrjagin dualHomtop-Ab.G;T / is contained in S, then G 2 ShAb S is dualizable.

Proof. By Lemma 3.5 we have isomorphisms

HomShAbS.G;T / Š Homtop-Ab.G;T /

and

HomShAbS.HomShAbS.G;T /;T / Š HomShAbS.Homtop-Ab.G;T /;T /

Š Homtop-Ab.Homtop-Ab.G;T /;T /:

The morphism c in (44) is induced by the evaluation map

G ! Homtop-Ab.Homtop-Ab.G;T /;T /

which is an isomorphism by the classical Pontrjagin duality of locally compact abeliangroups [Fol95], [HM98].

5.1.3 The sheaf of abelian groups T 2 ShAb S gives rise to a Picard stack BT 2 PIC.S/as explained in 2.3.3. Recall from 2.5.11 the following alternative description of BT .Let T Œ1� be the complex with the only non-trivial entry T Œ1��1 WD T . Then we haveBT Š ch.T Œ1�/ in the notation of 2.12.


5.1.4 Let now P 2 PIC.S/ be a Picard stack. Recall Definition 2.11, where we definethe internal HOM between two Picard stacks.

Definition 5.4. We define the dual stack by

D.P / WD HOMPIC.S/.P;BT /:

We hope that using the same symbolD for the dual in the case of Picard stacks andthe case of a sheaf of abelian groups will not cause confusion.

5.1.5 This definition is compatible with Definition 5.1 of the dual of a sheaf of abeliangroups in the following sense.

Lemma 5.5. If F 2 ShAb S, then we have a natural isomorphism

ch.D.F // Š D.BF /:Proof. First observe that by definitionD.BF / D HOMPIC.S/.ch.F Œ1�/; ch.T Œ1�//. Weuse Lemma 2.18 in order to calculate H i .D.BF //. We have

H�1.D.BF // Š R�1HomShAb S.F Œ1�;T Œ1�/ Š 0and

H 0.D.BF // Š R0HomShAb S.F Œ1�;T Œ1�/ Š HomShAb S.F;T / Š D.F /:

The composition of this isomorphism with the projectionD.BF /! ch.H 0.D.BF ///

from the stack D.BF / onto its sheaf of isomorphism classes (considered as Picardstack) provides the asserted natural isomorphism.

5.1.6 A sheaf of groups F 2 ShAb S can also be considered as a complex F 2C.ShAb S/ with non-trivial entry F 0 WD F . It thus gives rise to a Picard stack ch.F /.Recall the definition of an admissible sheaf 4.1.

Lemma 5.6. We have natural isomorphisms

H�1.D.ch.F /// Š D.F /; H 0.D.ch.F /// Š Ext1ShAb S.F;T /:

In particular, if F is admissible, then D.ch.F // Š B.D.F //.

Proof. We use again Lemma 2.18. We have

H�1.D.ch.F /// Š R�1HomShAb S.F;T Œ1�/ Š HomShAb S.F;T / Š D.F /:Furthermore,

H 0.D.ch.F /// Š R0HomShAb S.F;T Œ1�/ Š Ext1ShAb S.F;T /:


5.1.7 Assume now that F is dualizable and admissible (at this point we only needExt1ShAb S.F;T / Š 0). Then we have

D.D.ch.F /// Š D.B.D.F /// (by Lemma 5.6)

Š ch.D.D.F // (by Lemma 5.5)

Š ch.F /:

Similarly, if F is dualizable and D.F / admissible (again we only need the weakercondition that Ext1ShAb S.D.F /;T / Š 0), then we have, again by Lemmas 5.5 and 5.6,

D.D.BF // Š D.ch.D.F /// Š BD.D.F // Š BF:

5.1.8 Let us now formalize this observation. Let P 2 PIC.S/ be a Picard stack.

Definition 5.7. We callP dualizable if the natural evaluation morphismP ! D.D.P //

is an isomorphism.

The discussion of 5.1.7 can now be formulated as follows.

Corollary 5.8. 1. If F is dualizable and admissible, then ch.F / is dualizable.2. If F is dualizable and D.F / is admissible, then BF is dualizable.

The goal of the present subsection is to extend this kind of result to more generalPicard stacks.

5.1.9 Let P 2 PIC.S/.

Lemma 5.9. We haveH�1.D.P // Š D.H 0.P //

and an exact sequence

0! Ext1ShAb S.H0.P /;T /! H 0.D.P //! D.H�1.P //! Ext2ShAb S.H

0.P /;T /:(45)

Proof. By Lemma 2.13 we can chooseK 2 C.ShAb S/ such that P Š ch.K/. We nowget

H�1.D.P // Š R�1HomShAb S.K;T Œ1�/ (by Lemma 2.18)

Š R0HomShAb S.K;T /

Š HomShAb S.H0.K/;T /

Š D.H 0.P //:

Again by Lemma 2.18 we have

H 0.D.P // Š R0HomShAb S.K;T Œ1�/ Š R1HomShAb S.K;T /:


In order to calculate this sheaf in terms of the cohomology sheaves H i .K/ of K wechoose an injective resolution T ! I . Then we have

RHomShAb S.K;T / Š HomShAb S.K; I /:

We must calculate the first total cohomology of this double complex. We first take thecohomology in the K-direction, and then in the I -direction. The second page of theassociated spectral sequence has the following form.

1 Ext0ShAb S.H�1.P /;T/ Ext1ShAb S.H

�1.P /;T/ Ext2ShAb S.H�1.P /;T/ Ext3ShAb S.H

�1.P /;T/

0 Ext0ShAb S.H0.P /;T/ Ext1ShAb S.H

0.P /;T/ Ext2ShAb S.H0.P /;T/ Ext3ShAb S.H

0.P /;T/

0 1 2 3

The sequence (45) is exactly the edge sequence for the total degree-1 term.

5.1.10 The appearance in (45) of the groups ExtiShAb S.H0.P /;T / for i D 1; 2was the

motivation for the introduction of the notion of an admissible sheaf in 4.1.

Corollary 5.10. Let P 2 PIC.S/ be such that H 0.P / is admissible. Then we have

H 0.D.P // Š D.H�1.P //; H�1.D.P // Š D.H 0.P //:

Theorem 5.11. Let P 2 PIC.S/ and assume that

1. H 0.P / and H�1.P / are dualizable;

2. H 0.P / and D.H�1.P // are admissible.

Then P is dualizable.

Proof. In view of 2.14 it suffices to show that the evaluation map c W P ! D.D.P //

induces isomorphisms

H i .c/ W H i .P /! H i .D.D.P ///

for i D �1; 0. Consider first the case i D 0. Then by Corollary 5.10 we have anisomorphism

h0 W H 0.D.D.P ///�! D.H�1.D.F // �! D.D.H 0.P ///:

One now checks that the map

h0 ıH 0.c/ W H 0.P /! D.D.H 0.P ///

is the evaluation map (44). Since we assume that H 0.P / is dualizable this map is anisomorphism. Hence H 0.c/ is an isomorphism, too.

For i D �1 we use the isomorphism

h�1 W H�1.D.D.P /// �! D.H 0.D.P ///�! D.D.H�1.P ///

and the fact that h�1 ıH�1.c/ W H�1.P /! D.D.H�1.P /// is an isomorphism.


5.1.11 IfG is a locally compact group which together with its Pontrjagin dual belongsto S, then by 5.3 the sheaf G is dualizable. By Theorem 4.8 we know a large class oflocally compact groups which are admissible on S or at least after restriction to Slc orSlc-acyc.

We get the following result.

Theorem 5.12. Let G0; G�1 2 S be two locally compact abelian groups. We assumethat their Pontrjagin duals belong to S, and thatG0 andDG�1 are admissible on S. IfP 2 PIC.S/ has H i .P / Š Gi for i D �1; 0, then P is dualizable.

Let us specialize to the case which is important for the application to T -duality.Note that a group of the form Tn � Rm � F for a finitely generated abelian group Fis admissible by Theorem 4.8. This class of groups is closed under forming Pontrjaginduals.

Corollary 5.13. If P 2 PIC.S/ has H i .P / Š Tni � Rmi � Fi for some finitelygenerated abelian groups Fi for i D �1; 0, then P is dualizable.

For more general groups one may have to restrict to the sub-site Slc or even toSlc-acyc.

5.2 Duality and classification

5.2.1 Let A;B 2 ShAb S. By Lemma 2.20 we know that the isomorphism classesExtPIC.S/.A;B/ of Picard stacks withH 0.P / Š B andH�1.P / Š A are classified bya characteristic class

� W ExtPIC.S/.B;A/�! Ext2ShAb S.B;A/: (46)

Lemma 5.14. If A is dualizable and D.A/;B are admissible, then there is a naturalisomorphism

D W Ext2ShAb S.B;A/�! Ext2ShAb S.D.A/;D.B//

such that

�.D.P // D D.�.P // for all P 2 ExtPIC.S/.A;B/: (47)

Proof. In order to define D we use the identifications

Ext2ShAb S.B;A/ Š HomDC.ShAb S/.B;AŒ2�/;

Ext2ShAb S.D.A/;D.B// Š HomDC.ShAb S/.D.A/;D.B/Œ2�/:

We choose an injective resolution T ! I. For a complex of sheaves F we defineRD.F / WD HomShAb S.F; I/. The map T ! I induces a map D.F / ! RD.F /.


Note thatRD descends to a functor between derived categoriesRD W Db.ShAb S/op !DC.ShAb S/. We now consider the following web of maps:

HomDC.ShAb S/.B;AŒ2�/

D

��

D 0

RD �� HomDC.ShAb S/.RD.AŒ�2�/; RD.B//D.A/!RD.A/��

HomDC.ShAb S/.D.A/;RD.B/Œ2�/

HomDC.ShAb S/.D.A/;D.B/Œ2�/.

u D.B/!RD.B/��

Since B is admissible the map D.B/ ! RD.B/ is an isomorphism in cohomologyin degree 0; 1; 2 (because D.B/ is concentrated in degree 0 and H k.RD.B// DRkHomShAb S.B;T / D ExtkShAb S.B;T /). Since D.A/ is acyclic in non-zero degree,the map u is an isomorphism. For this, observe thatD.B/! RD.B/ can be replacedup to quasi-isomorphism by an embedding 0 ! BRD.B/ ! RD.B/ ! Q ! 0

such that the quotient is zero in degrees 0; 1; 2. The statement then follows from thelong exact Ext-sequence for HomShAb S.D.A/;�/, because ExtiShAb S.D.A/; B/ D 0 fori D 0; 1; 2. Therefore the diagram defines the map

D WD u�1 ıD 0 W HomDC.ShAb S/.B;AŒ2�/! HomDC.ShAb S/.D.A/;D.B/Œ2�/:

5.2.2 We now show the relation (47). By Lemma 2.13 it suffices to show (47) forP 2 PIC.S/ of the form P D ch.K/ for complexes

K W 0! A! X ! Y ! B ! 0; K W 0! X ! Y ! 0:

As in 2.5.9 we consider

KA W 0! X ! Y ! B ! 0

with B in degree 0. Then by Definition (12) of the map � and the Yoneda map Y , theelement �.ch.K// 2 HomDC.ShAb S/.B;AŒ2�/ is represented by the composition (see(10))

Y.K/ W B ˇ!KA˛ � AŒ2�:

Since RD preserves quasi-isomorphisms we get RD.˛�1/ D RD.˛/�1. It followsthat

RD.Y.K// W RD.AŒ2�/ RD.˛/�1

��! RD.KA/RD.ˇ/��! RD.B/:

We read off that

D 0.Y.K// W D.A/! RD.A/RD.˛/�1��! RD.KA/Œ2�

RD.ˇ/��! RD.B/Œ2�:


Let T ! I 0 ! I 1 ! : : : be the injective resolution I of T . We define J WDker.I 1 ! I 2/ and I WD I 0. Then we have BT D ch.L/ with L W 0! I ! J ! 0

with J in degree zero. Note that I is injective. Then by Lemma 2.17 we have

D.P / Š ch.H/; (48)

where H WD ��0HomShAb S.K;L/. Let

Q WD ker.HomShAb S.X; I /˚ HomShAb S.Y; J /! HomShAb S.X; J //:

Then H is the complex

H W 0! HomShAb S.Y; I /! Q! 0:

There is a natural map D.B/ ! HomShAb S.Y; I / (induced by T ! I and Y ! B),and a projection Q ! D.A/ induced by A ! X and passage to cohomology. SinceB is admissible the complex

H W 0! D.B/! HomShAb S.Y; I /d�! Q! D.A/! 0

is exact. Note that ker.d/DH�1HomShAb S.K; I / and coker.d/DH 0HomShAb S.K; I /.We get

�.D.P //.48)D �.ch.H//

(12)D Y.H /Lemma 2.19D Y 0.H /:

Explicitly, in view of (11) the map Y 0.H / 2 HomDC.ShAb S/.D.A/;D.B/Œ2�/ is givenby the composition

Y 0.H / W D.A/ ��1

��! HD.A/

ı! D.B/Œ2�;

whereHD.A/ W 0! D.B/! HomShAb S.Y; I /! Q! 0

with Q in degree 0, the map W HD.A/ ! D.A/ is the quasi-isomorphism inducedby the projection Q ! D.A/, and ı W HD.A/ ! D.B/Œ2� is the canonical projection.SinceB is admissible we haveR1HomShAb S.B;T / Š 0 and hence a quasi-isomorphismHD.A/ ! H 0

D.A/fitting into the following larger diagram.

HD.A/ W

�

��

0 �� D.B/

��

�� HomShAb S.Y; I /

��

�� Q �� 0

H 0

D.A/W

��

0 �� HomShAb S.B; I /

��

��

HomShAb S.B; J /

˚HomShAb S.Y; I /

��

�� Q ��

��

0

RD.KA/ W 0 �� HomShAb S.B; I / ��

HomShAb S.B; I1/

˚HomShAb S.Y; I /

�� Hom2ShAb S.KA; I/�� : : :


5.2.3 The final step in the verification of (47) follows from a consideration of thediagram

D.B/Œ2� �� RD.B/Œ2�

HD.A/

� ��

ı

��

� �� H 0D.A/

��

��

RD.KA/

RD.˛/ ��

RD.ˇ/

��

D.A/

Y 0.H/

��uıY 0.H/

��

D.A/

D 0.Y.K//

��

�� RD.A/

RD.Y.K//

��

showing the marked equality in

�.D.P // D Y 0.H /ŠD D.Y.K// D D.�.P //:

It remains to show that

D W Ext2ShAb S.B;A/! Ext2ShAb S.D.A/;D.B//

is an isomorphism.

To this end we look at the following commutative web of maps

R2Hom ShAb S.D.A/;D.B//

uŠ

��

R2Hom ShAb S.B;A/D

��

D0

��

RD

��

z

��

Š �� R2Hom ShAb S.B;D.D.A///

Š v

��

R2Hom ShAb S.D.A/;RD.B//Š �� R2Hom ShAb S.D.A/˝L B;T/ R2Hom ShAb S.B;RD.D.A//

Š

��

R2Hom ShAb S.RD.A/;RD.B//

��

Š �� R2Hom ShAb S.RD.A/˝L B;T/

��

R2Hom ShAb S.B;RD.RD.A///.Š

��

��

The horizontal isomorphisms in the two lower rows are given by the derived adjointnessof the tensor product and the internal homomorphisms. The horizontal isomorphismin the first row is induced by the isomorphism A! D.D.A//. The maps u and v areisomorphisms since we assume that D.A/ is admissible, compare the correspondingargument in the proof of Lemma 5.14. The map z is induced by the canonical mapA! RD.RD.A//.


6 T -duality of twisted torus bundles

6.1 Pairs and topological T -duality

6.1.1 The goal of this subsection is to introduce the main objects of topological T -du-ality and review the structure of the theory. Let B be a topological space.

Definition 6.1. A pair .E;H/ over B consists of a locally trivial principal Tn-bundleE ! B and a gerbe H ! E with band TjE . An isomorphism of pairs is a diagram

H

��

�� H 0

��

E

��

�� E 0

��

B B

consisting of an isomorphism of Tn-principal bundles � and an isomorphism ofT -banded gerbes . By P.B/ we denote the set of isomorphism classes of pairsover B . If f W B ! B 0 is a continuous map, then pull-back induces a functorial mapP.f / W P.B 0/! P.B/.

6.1.2 The functorP W TOPop ! Sets

has been introduced in [BS05] for n D 1 and in [BRS] for n 1 in connection withthe study of topological T -duality.

The main result of [BS05] in the case n D 1 is the construction of an involutivenatural isomorphism T W P ! P , the T -duality isomorphism, which associates to eachisomorphism class of pairs t 2 P.B/ the class of the T -dual pair Ot WD T .t/ 2 P.B/.

In the higher-dimensional case n 2 the situation is more complicated. First ofall, not every pair t 2 P.B/ admits a T -dual. Furthermore, a T -dual, if it exists, maynot be unique. The results of [BRS] are based on the notion of a T -duality triple. Inthe following we recall the definition of a T -duality triple.

6.1.3 As a preparation we recall the following two facts. Let � W E ! B be aTn-principal bundle. Associated to the decomposition of the global section functor�.E; : : : / as composition �.E; : : : / D �.B; : : : / ı �� W ShAb S=E ! Ab there is adecreasing filtration

� F nH�.EIZ/ � F n�1H�.EIZ/ � � F 0H�.EIZ/of the cohomology groups H�.EIZ/ Š R��.EIZ/ and a Serre spectral sequencewith second term E

i;j2 Š H i .BIRj��Z/ which converges to GrH�.EIZ/.


6.1.4 The isomorphism classes of gerbes over X with band T jX are classified byH 2.X IT / Š H 3.X IZ/. The class associated to such a gerbe H ! X is calledDixmier–Douady class d.X/ 2 H 3.X IZ/.

If H ! X is a gerbe with band T jX over some space X , then the automorphismsof H (as a gerbe with band T jX ) are classified by H 1.X IT / Š H 2.X IZ/.6.1.5 In the definition of a T -duality triple we furthermore need the following notation.We let y 2 H 1.T IZ/ be the canonical generator. If pri W Tn ! T is the projectiononto the i th factor, then we set yi WD pr�i y 2 H 1.TnIZ/. Let E ! B be a Tn-principal bundle and b 2 B . We consider its fibre Eb over b. Choosing a base point

e 2 Eb we use the Tn-action in order to fix a homeomorphism ae W E �! Tn such thatae.et/ D t for all t 2 Tn. The classes xi WD a�e .yi / 2 H 1.EbIZ/ are independentof the choice of the base point. Applying this definition to the bundle yE ! B belowgives the classes Oxi 2 H 1. yEbIZ/ used in Definition 6.2.

Definition 6.2. A T -duality triple t WD ..E;H/; . yE; yH/; u/ over B consists of twopairs .E;H/; . yE; yH/ over B and an isomorphism u W Op� yH ! p�H of gerbes withband T jE�B yE defined by the diagram

p�H

��

��

�

�� Op� yHu

��

��

��

H

��

��

��E �B yE

p

�� Op

��

yH

��

��

�

E

�

�� yE

O��

B ,

(49)

where all squares are two-cartesian. The following conditions are required:

1. The Dixmier–Douady classes of the gerbes satisfy d.H/ 2 F 2H 3.EIZ/ andd. yH/ 2 F 2H 3. yEIZ/.

2. The isomorphism of gerbes u satisfies the condition [BRS, (2.7)] which says thefollowing. If we restrict the diagram (49) to a point b 2 B , then we can trivializethe restrictions of gerbes HjEb , yHj yEb to the fibres Eb; yEb of the Tn-bundles

over b such that the induced isomorphism of trivial gerbes ub over Eb � yEb isclassified by

PniD1 pr�Ebxi [ pr�yE Oxi 2 H 2.Eb � yEbIZ/.

6.1.6 There is a natural notion of an isomorphism of T -duality triples. For a mapf W B 0 ! B and a T -duality triple over B there is a natural construction of a pull-backtriple over B 0.


Definition 6.3. We let Triple.B/ denote the set of isomorphism classes of T -dualitytriples over B . For f W B 0 ! B we let Triple.f / W Triple.B/! Triple.B 0/ be the mapinduced by the pull-back of T -duality triples.

6.1.7 In this way we define a functor

Triple W TOPop ! Sets:

This functor comes with specializations

s; Os W Triple! P

given by s..E;H/; . yE; yH/; u/ WD .E;H/ and Os..E;H/; . yE; yH/; u/ D . yE; yH/.Definition 6.4. A pair .E;H/ is called dualizable if there exists a triple t 2 Triple.B/such that s.t/ Š .E;H/. The pair Os.t/ D . yE; yH/ is called a T -dual of .E;H/.

Thus the choice of a triple t 2 s�1.E;H/ encodes the necessary choices in orderto fix a T -dual. One of the main results of [BRS] is the following characterization ofdualizable pairs.

Theorem 6.5. A pair .E;H/ is dualizable in the sense of Definition 6.4 if and only ifd.H/ 2 F 2H 3.EIZ/.

Further results of [BRS, Theorem 2.24] concern the classification of the set of dualsof a given pair .E;H/.

6.1.8 For the purpose of the present paper it is more natural to interpret the isomorphismof gerbes u W Op� yH ! p�H in a T -duality triple ..E;H/; . yE; yH/; u/ in a different,but equivalent manner. To this end we introduce the notion of a dual gerbe.

6.1.9 First we recall the definition of the tensor product w W H ˝X H 0 ! X of gerbesu W H ! X and u0 W H 0 ! X with band T jX over X 2 S. Consider first the fibreproduct of stacks .u; u0/ W H �X H 0 ! X . Let T 2 S=X . An object s 2 H �X H 0.T /is given by a triple .t; t 0; �/ of objects t 2 H.T /, t 0 2 H 0.T / and an isomorphism� W u.t/! u0.t 0/. By the definition of a TjX -banded gerbe the group of automorphismsof t relative to u is the group AutH.T /=rel.u/.t/ Š T .T /. We thus have an isomorphismAutH�XH 0=rel..u;u0//.s/ Š T .T / � T .T /. Similarly, for s0; s1 2 H �X H 0.T / the setHomH�XH 0=rel..u;u0//.s0; s1/ is a torsor over T .T / � T .T /.

We now define a prestackH˝pXH 0. By definition the groupoidH˝pXH 0.T / has thesame objects as H �X H 0.T /, but the morphism sets are factored by the anti-diagonal

T .T /antidiag� T .T / � T .T /, i.e.

HomH˝pXH 0=rel.w/.s0; s1/ D HomH�XH 0=rel..u;u0//.s0; s1/=antidiag.T .T //:

The stackH˝XH 0 is defined as the stackification of the prestackH˝pXH 0. It is again agerbe with band T jX . We furthermore have the following relation of Dixmier–Douadyclasses.

d.H/C d.H 0/ D d.H ˝X H 0/:


6.1.10 The sheaf T jX 2 ShAb S=X gives rise to the stack BT jX (see 2.3.6). ThenX �BT jX ! X is the trivial T -banded gerbe over X . Let H ! X be a gerbe withband T jX .

Definition 6.6. A dual of the gerbeH ! X is a pair .H 0 ! X; / of a gerbeH 0 ! X

and an isomorphism of gerbes W H ˝X H 0 ! X �BT jX .

Every gerbe H ! X with band T jX admits a preferred dual H op ! X which wecall the opposite gerbe. The underlying stack of H op is H , but we change its structure

of a T jX -banded gerbe using the inversion automorphism �1 W T jX�! T jX .

If .H 00 ! X; 0/ and .H 01 ! X; / are two duals, then there exists a uniqueisomorphism class of isomorphismsH 00 ! H 01 of gerbes such that the induced diagram

H ˝X H 00 0

�� H ˝X H 01

1��

X �BT jX

can be filled with a two-isomorphism. Note that ifH 0 ! X is underlying gerbe of thedual of H ! X , then we have the relation of Dixmier–Douady classes

d.H/ D �d.H 0/:

Lemma 6.7. Let yH ! X and H ! X be gerbes with band T jX . There is a natural

bijection between the sets of isomorphism classes of isomorphisms yH ! H of gerbesoverX and isomorphism classes of isomorphisms of T jX -banded gerbes yH ˝H op !X �BT jX .

Proof. An isomorphism yH ! H induces an isomorphism yH˝XH op ! H˝XH op !X �BT jX . On the other hand, an isomorphism yH ˝X H op ! X �BT jX presentsyH ! X as a dual of H op ! X . Since W H ˝X H op ! X �BT jX presents H as

a dual of H op we have a preferred isomorphism class of isomorphisms yH ! H , too.

6.1.11 In view of Lemma 6.7, in Definition 6.2 of aT -duality triple ..E;H/; . yE; yH/; u/we can consider u as an isomorphism

Op� yH op ˝X p�H ! E �B yE �BT jE�B yE :

The condition 6.2 2. can be rephrased as follows. After restriction to a point b 2 B wecan find isomorphisms

v W Tn �BT�! Hb; Ov W Tn �BT

�! yHb: (50)


After a choice of � ! BT in order to define the map s below we obtain a mapr W Tn � Tn ! BT by the following diagram.

yH opb�Hb �� yH op

b˝Eb� yEb Hb

ub �� Eb � yEb �BTprBT

�� BT

.Tn �BT op/ � .Tn �BT /

. Ov;v/��

Tn � Tn

r

��

s

��

The condition 6.2, 2. is now equivalent to the condition that we can choose the isomor-phisms v; Ov in (50) such that

r�.z/ DnXiD1

pr�1;ix [ pr�2;ix;

where z 2 H 2.BT IZ/ and x 2 H 1.T IZ/ are the canonical generators, and

prk;i ; W Tn � Tn prk��! Tnpri�! T , k D 1; 2, i D 1; : : : ; n are projections onto the

factors.

6.1.12 Important topological invariants of T -duality triples are the Chern classes ofthe underlying Tn-principal bundles. For a triple t D ..E;H/; . yE; yH/; u/ we define

c.t/ WD c.E/; Oc.t/ WD c. yE/:These classes belong to H 2.BIZn/.

6.2 Torus bundles, torsors and gerbes

6.2.1 In this subsection we review various interpretations of the notion of a Tn-prin-cipal bundle.

6.2.2 Let G be a topological group. Let us start with giving a precise definition of aG-principal bundle.

Definition 6.8. A G-principal bundle E over a space B consists of a map of spaces� W E ! B which admits local sections together with a fibrewise right actionE�G !E such that the natural map

E �G ! E �B E; .e; t/ 7! .e; et/

is an homeomorphism. An isomorphism of G-principal bundles E ! E 0 is a G-equi-variant map E ! E 0 of spaces over B .


By PrinB.G/ we denote the category of G-principal bundles over B . The setH 0.PrinB.G// of isomorphism classes in PrinB.G/ is in one-to-one correspondencewith homotopy classes ŒB;BG � of maps from B to the classifying space BG of G. Ifwe fix a universal bundle EG! BG, then the bijection

ŒB;BG ��! H 0.PrinB.G//

is given byŒf W B ! BG � 7! ŒB �f;BG EG! B�:

6.2.3 We now specialize to G WD Tn. The classifying space BTn of Tn has thehomotopy type of the Eilenberg–MacLane space K.Zn; 2/. We thus have a naturalisomorphism

H 2.BIZn/ Def.Š ŒB;K.Zn; 2/� Š ŒB; BTn� Š H 0.PrinB.Tn//:

So, Tn-principal bundles are classified by the characteristic class c.E/ 2 H 2.BIZn/.Using the decomposition

H 2.BIZn/ Š H 2.BIZ/˚ : : :H 2.BIZ/„ ƒ‚ …n summands

we can writec.E/ D .c1.E/; : : : ; cn.E//:

Definition 6.9. The class c.E/ is called the Chern class of E . The ci .E/ are called thecomponents of c.E/.

In fact, if n D 1, then c.E/ is the classical first Chern class of the T -principalbundle E .

6.2.4 Let S be a site and F 2 ShAbS be a sheaf of abelian groups.

Definition 6.10. An F -torsor T is a sheaf of sets T 2 ShS together with an actionT �F ! T such that the natural map T �F ! T �T is an isomorphism of sheaves.An isomorphism of F -torsors T ! T 0 is an isomorphism of sheaves which commuteswith the action of F .

By Tors.F / we denote the category of F -torsors.

6.2.5 In 2.3.4 we have introduced the category EXT.Z; F /whose objects are extensionsof sheaves of groups

W W 0! F ! Ww! Z! 0; (51)

and whose morphisms are isomorphisms of extensions. We have furthermore definedthe equivalence of categories

U W EXT.Z; F /�! Tors.F /

given by U.W/ WD w�1.1/.


6.2.6 Consider an extension W as in (51) and apply ExtShAb S.Z; : : : /. We get thefollowing piece of the long exact sequence

! HomShAb S.Z; W /! HomShAb S.Z;Z/

ıW��! Ext1ShAb S.Z; F /! Ext1ShAb S.Z; W /! :

Let 1 2 HomShAb S.Z;Z/ be the identity and set

e.W/ WD ıW .1/ 2 Ext1ShAb S.Z; F /:

Recall that H 0.EXT.Z; F // denotes the set of isomorphism classes of the categoryEXT.Z; F /. The following lemma is well-known (see e.g. [Yon60]).

Lemma 6.11. The map

EXT.Z; F / 3 W 7! e.W/ 2 Ext1ShAb S.Z; F /

induces a bijection

e W H 0.EXT.Z; F //�! Ext1ShAb S.Z; F /:

In view of 6.2.5 we have natural bijections

H 0.Tors.F //UŠ H 0.EXT.Z; F //

eŠ Ext1ShAb S.Z; F /: (52)

6.2.7 Let G be an abelian topological group and consider a principal G-bundle E

over B with underlying map � W E ! B . By T .E/ WD E ! B 2 ShS=B (see3.2.3) we denote its sheaf of sections. The right action of G on E induces an actionT .E/ �GjB ! T .E/.

Lemma 6.12. T .E/ is a GjB -torsor.

Proof. This follows from the following fact. If X ! B and Y ! B are two maps,then

X �B Y ! B Š X ! B � Y ! B

in ShS=B . We apply this to the isomorphism

E �B .B �G/ Š E �G Š E �B E

of spaces over B in order to get the isomorphism

T .E/ �GjB�! T .E/ � T .E/:


6.2.8 The construction E 7! T .E/ refines to a functor

T W PrinB.G/! Tors.GjB/

from the category of G-principal bundles PrinB.G/ over B to the category of GjB -torsors Tors.GjB/ over B .

Lemma 6.13. The functor

T W PrinB.G/! Tors.GjB/

is an equivalence of categories.

Proof. It is a consequence of the Yoneda Lemma that T is an isomorphism on the levelof morphism sets. It remains to show that the underlying sheaf T of a GjB -torsor isrepresentable by a G-principal bundle. Since T is locally isomorphic to GjB this istrue locally. The local representing objects can be glued to a global representing object.

We can now prolong the chain of bijections (52) to

ŒB;BG� Š H 0.PrinB.G// Š H 0.Tors.GjB//Š H 0.EXT.ZjB ; GjB// Š Ext1ShAb S=B.ZjB ; GjB/ Š H 1.BIG/: (53)

6.2.9 Let S be some site and F 2 S be a sheaf of abelian groups. By Gerbe.F /we denote the two-category of gerbes with band F over S. It is well-known thatisomorphism classes of gerbes with band F are classified by Ext2ShAb S.ZIF /, i.e. thereis a natural bijection

d W H 0.Gerbe.F //�! Ext2ShAb S.ZIF /:

6.2.10 Let H ! G be a homomorphism of topological abelian groups with kernelK WD ker.H ! G/. Our main example is Rn ! Tn with kernel Zn.

Definition 6.14. Let T be a space. An H -reduction of a G-principal bundle E D.E ! B/ on T is a diagram

.F; �/ W F ��

��

E

��

T �� B

;

where F ! T is an H -principal bundle, and � is H -equivariant, where H acts onE via H ! G. An isomorphism .F; �/ ! .F 0; �0/ of H -reductions over T is anisomorphism f W F ! F 0 of H -principal bundles such that �0 ı f D �. Let RE

H .T /

denote the category of H -reductions of E .


Observe that the group of automorphisms of every object in REH .T / is isomorphic

to Map.T;K/ı (the superscript ı indicates that we take the underlying set). For a mapu W T ! T 0 over B there is a natural pull-back functor RE

H .u/ W REH .T

0/! REH .T /.

Let S be a site as in 3.1.2 and assume thatK;G;H;B belong to S. In this case it iseasy to check that

T 7! REH .T /

is a gerbe with band K jB on S=B .

6.2.11 Let� W E ! B be aG-principal bundle. Note that the pull-back��E ! E hasa canonical section and is therefore trivialized. A trivialized G-bundle has a canonicalH -reduction. In other words, there is a canonical map of stacks over B

can W E ! REH : (54)

Note that an object of E.T / is a map T ! E; to this we assign the pull-back of thecanonical H -reduction of ��E.

6.2.12 The construction PrinB.G/ 3 E 7! REH 2 Gerbe.K/ is functorial in E and

thus induces a natural map of sets of isomorphism classes

r W H 0.PrinB.G//! H 0.Gerbe.K//:

Lemma 6.15. If H ! G is surjective and has local sections, and H 1.B;H/ ŠH 2.B;H/ Š 0, then

r W H 0.PrinB.G//! H 0.Gerbe.K//

is a bijection.

Proof. The exact sequence 0! K ! H ! G ! 0 induces by 3.4 an exact sequence

0! K ! H ! G ! 0

of sheaves. We consider the following segment of the associated long exact sequencein cohomology:

! H 1.BIH/! H 1.BIG/ ı! H 2.BIK/! H 2.BIH/! :By our assumptions ı W H 1.BIG/ ! H 2.BIK/ is an isomorphism. One can checkthat the following diagram commutes:

H 0.PrinB.G//r ��

��

H 0.Gerbe.K jB//

d

��

H 1.BIG/ ı �� H 2.BIK/.(55)

This implies the result since the vertical maps are isomorphisms.


Note that we can apply this lemma in our main example where H D Rn andG D Tn. In this case the diagram (55) is the equality

c.E/ D d.REZnjB/ (56)

in H 2.BIZn/.

6.3 Pairs and group stacks

6.3.1 Let E be a principal Tn-bundle over B , or equivalently by (53), an extensionE 2 ShAb S=B

0! TnjB ! E ! ZjB ! 0 (57)

of sheaves of abelian groups. Let Qc.E/ 2 Ext1ShAb S=B.ZjB ITnjB/ be the class of this

extension. Under the isomorphism

Ext1ShAb S=B.ZjB ITnjB/ Š H 1.BITn/ Š H 2.BIZn/

it corresponds to the Chern class c.E/ of the principal Tn-bundle introduced in 6.9.

6.3.2 We let QE D ExtPIC.S/.E;T jB/ (see Lemma 2.20 for the notation) denote theset of equivalence classes of Picard stacks P 2 PIC.B/ with isomorphisms

H 0.P /Š�! E; H�1.P / Š�! TB :

By Lemma 2.20 we have a bijection

Ext2ShAb S=B.E;T jB/ Š QE :

This bijection induces a group structure on QE which we will use in the discussion oflong exact sequences below. We will not need a description of this group structure interms of the Picard stacks themselves.

We apply Ext�ShAb S=B.: : : ;T jB/ to the sequence (57) and get the following segmentof a long exact sequence

Ext1ShAb S=B.TnjB ;T jB/

˛! Ext2ShAb S=B.ZjB ;T jB/! QE

! Ext2ShAb S=B.TnjB ;T jB/

ˇ! Ext3ShAb S=B.ZjB ;T jB/:(58)

The maps ˛; ˇ are given by the left Yoneda product with the class

Qc.E/ 2 Ext1ShAb S=B.ZjB ITnjB/:


6.3.3 In this paragraph we identify the extension groups in the sequence (58) with sheafcohomology. For a site S with final object � and X; Y 2 ShAb S we have a local-globalspectral sequence with second term

Ep;q2 Š Rp�.�IExtqShAb S.X; Y //

which converges to ExtpCqShAb S.X; Y /.We apply this first to the sheaves ZjB ;T jB 2 ShAbS=B . The final object of the site

S=B is id W B ! B . We have ExtqShAb S=B.ZjB ;T jB/ Š 0 for q 1 so that this spectralsequence degenerates at the second page and gives

ExtpShAb S=B.ZjB ;T jB/ Š Hp.BIHomShAb S=B.ZjB ;T jB//

Š Hp.BIT jB/ Š Hp.BIT / Š HpC1.BIZ/:Since the group Tn is admissible by Theorem 4.30, and by Corollary 3.14 we have

ExtqShAb S=B.TnjB ;T jB/ Š 0 for q D 1; 2, we get

ExtpShAb S=B.TnjB ;T jB/ Š Hp.BIHomShAb S=B.T

njB ;T jB//

Š Hp.BIHomShAb S.Tn;T /jB/

Š Hp.BIZnjB/Š Hp.BIZn/

for p D 1; 2. Therefore the sequence (58) has the form

H 1.BIZn/ ˛! H 3.BIZ/! QE

Oc! H 2.BIZn/ ˇ! H 4.BIZ/: (59)

In this picture the maps ˛; ˇ are both given by the cup-product with the Chern classc.E/ 2 H 2.BIZn/, i.e. ˛..xi // DP

xi [ ci .E/.6.3.4 Recall from 6.1.3 the decreasing filtration .F kH�.EIZ//k�0 and the spec-tral sequence associated to the decomposition of functors �.EI : : : / D �.B; : : : / ıp� W ShAbS=E ! Ab. This spectral sequence converges to GrH�.EIZ/. Its secondpage is given byEi;j2 Š H i .BIRjp�Z/. The edge sequence for F 2H 3.EIZ/ has theform

ker.d0;22 W E0;22 ! E2;12 /

d1;13��! coker.d1;12 W E1;12 ! E

3;02 /

! F 2H 3.EIZ/! .E2;12 =im.d0;22 W E0;22 ! E

2;12 //

d2;12��! E

4;02 :

We now make this explicit. Since the fibre of p is an n-torus, R0p�Z Š Z,R1p�Z Š Zn, R2p�Z Š ƒ2Zn. The differential d2 can be expressed in terms of theChern class c.E/. We get the following web of exact sequences, where K, A, B aredefined as the appropriate kernels and cokernels.


KD

��

��

H 1.BIZn/˛

��

H 0.BIƒ2Zn/ic.E/

��

H 3.BIZ/

��

H 2.BIZn/ˇ

��

��

K

s

��

�� A ��

��

F 2H 3.EIZ/ �� BN

��

��

H 4.BIZ/

0 0

(60)

Since K as a subgroup of the free abelian group H 0.B;ƒ2Zn/ is free we can choosea lift s as indicated.

6.3.5 We now define a map (the underlying pair map)

up W QE ! P.B/

as follows. Any Picard stack P 2 PIC.S=B/ comes with a natural map of stacksP ! H 0.P / (where on the right-hand side we consider a sheaf of sets as a stack). IfP 2 QE , then H 0.P / Š E has a natural map to ZjB (see (57)). We define the stackG by the pull-back in stacks on S=B

G

��

��

�� P

��

��

E ��

��

E

��¹1ºjB �� ZjB .

All squares are two-cartesian, and the outer square is the composition of the two innersquares. We have omitted to write the canonical two-isomorphisms. By constructionE is a sheaf of Tn-torsors (see 6.2.5), i.e. by Lemma 6.13 a Tn-principal bundle, andG ! E is a gerbe with band T . We set

up.P / WD .E;G/:


6.3.6 Recall Definition 6.4 of a dualizable pair.

Lemma 6.16. If P 2 QE , then up.P / 2 P.B/ is dualizable.

Proof. Let .E;H/ WD up.E/. In view of Theorem 6.5 we must show that d.H/ 2F 2H 3.EIZ/. For k 2 Z we define G.k/! E.k/ by the two-cartesian diagrams

G.k/

��

�� P

!!

��

E.k/ ��

��

E

��¹kºjB �� ZjB .

(61)

The group structure of E induces maps

� W E.k/ �B E.m/! E.k Cm/ (62)

On fibres appropriately identified with Tn, this map is the usual group structure on Tn.Since P is a Picard stack these multiplications are covered by O� W G.k/ � G.m/ !G.k Cm/. The isomorphism class of G.k/ therefore must satisfy

pr�E.k/G.k/˝ pr�E.m/G.m/ Š ��G.k Cm/ 2 Gerbe.E.k/ �B E.m//: (63)

We now write out this isomorphism in terms of Dixmier–Douady classes dk WDd.G.k// 2 H 3.E.k/IZ/.

We fix a generator ofH 1.T IZ/. This fixes a choice of generators ofxi 2H 1.TnIZ/,i D 1; : : : ; n via pullback along the coordinate projections. Let a W Tn �Tn ! Tn bethe group structure. Then we have

a�.xi / D pr�1xi C pr�2xi : (64)

where pri W Tn � Tn ! Tn, i D 1; 2, are the projections onto the factors.Let us for simplicity assume that B is connected. Let .Er.k/; dr.k// be the Serre

spectral sequence of the compositionE.k/! B ! � (see 6.1.7). Then we can identifyE0;32 .k/ Š ƒ3ZH 1.TnIZ/. The class dk has a symbol inE0;32 .k/which can be written

asPi<j<l ai;j;l.k/xi ^ xj ^ xl . Since the map � (see 62) on fibres can be written in

terms of the group structure we can use (64) in order to write out the symbol of��.dk/.Equation (63) implies the identity

Xi;j;l

ai;j;l.k/pr�1.xi ^ xj ^ xl/CXs;t;u

as;t;u.m/pr�2.xs ^ xt ^ xu/

DXa;b;c

aa;b;c.k Cm/.pr�1xa C pr�2xa/ ^ .pr�1xb C pr�2xb/ ^ .pr�1xc C pr�2xc/:


Because of the presence of mixed terms this is only possible if everything vanishes.This implies that dk 2 F 1H 3.E.k/IZ/. We write

Pi;j xi^xj˝ui;j .k/ for its symbol

in E1;22 , where ui;j 2 H 1.BIZ/. As above equation (63) now implies

Xi;j

pr�1.xi ^ xj /˝ ui;j .k/CXl;k

pr�2.xl ^ xr/˝ ul;r.m/

DXa;b

.pr�1xa C pr�2xa/ ^ .pr�1xb C pr�2xb/˝ ua;b.k Cm/:

Again the presence of mixed terms implies that everything vanishes. This shows thatdk 2 F 2H 3.E.k/IZ/. The assertion of the lemma is the case k D 1.

6.3.7 Let us now combine (60) and (59) into a single diagram. We get the followingweb of horizontal and vertical exact sequences:

H 1.BIZn/

��

��˛

��

H 1.BIZn/ �� H 3.BIZ/ h ��

��

QE

f

��

Oc �� H 2.BIZn/

��

ˇ�� H 4.BIZ/

K ��

s

""

A ��

��

F 2H 3.EIZ/

��

�� B

��

N�� H 4.BIZ/

0 0 0.(65)

The map f W QE ! F 2H 3.EIZ/ associates to P 2 QE the Dixmier–Douady classof the gerbe of the pair up.P / 2 P.B/ with underlying Tn-bundle E. Here we useLemma 6.16. Surjectivity of f follows by a diagram chase once we have shown thefollowing lemma.

Lemma 6.17. The diagram (65) commutes.

Proof. We have to check that the left and the right squares

H 3.BIZ/

��

h �� QE

f

��

A �� F 2H 3.EIZ/and

QEOc ��

f

��

H 2.BIZn/

��

F 2H 3.EIZ/ e �� B

(66)

commute. Let us start with the left square. Let d 2 H 3.BIZ/. Under the identification

H 3.BIZ/ Š H 2.BIT / Š Ext2ShAb S=B.ZIT /


it corresponds to a group stack P 2 PIC.S=B/ with H 0.P / Š Z and H�1.P / Š T .The group stack h.d/ 2 QE is given by the pull-back (two-cartesian diagram)

h.d/

��

�� E

��

P �� Z.

In particular we see that the gerbe H ! E of the pair u.h.d// DW .E;H/ is given bya pull-back

H

��

�� E

��

G �� B ,

where G 2 Gerbe.B/ is a gerbe with Dixmier–Douady class d.G/ D d . The com-position f ı h W H 3.BIZ/! H 3.EIZ/ is thus given by the pull-back along the mapp W E ! B , i.e. p� D f ı h. By construction the composition H 3.BIZ/ ! A !F 2H 3.EIZ/ is a certain factorization ofp�. This shows that the left square commutes.

Now we show that the right square in (66) commutes. We start with an explicitdescription of Oc. Let P 2 QE . The principal Tn-bundle in G.0/ ! E.0/ ! B istrivial (see (61) for notation). Therefore the Serre spectral sequence .Er.0/; dr.0//degenerates at the second term. We already know by Lemma 6.16 that the Dixmier–Douady class of G.0/ ! E.0/ satisfies d0 2 F 2H 3.E.0/IZ/. Its symbol can bewritten as

PniD1 xi ˝ Oci .P / for a uniquely determined sequence Oci .P / 2 H 2.BIZ/.

These classes constitute the components of the class Oc.P / 2 H 2.BIZn/.We write the symbol of f .P / D d1 as

Pi xi ˝ ai for a sequence ai 2 H 2.BIZ/.

As in the proof of Lemma 6.16 the equation (63) gives the identity

nXiD1

pr�2xi ˝ ai CnXiD1

pr�1xi ˝ Oci .P / �nXiD1.pr�1xi C pr�2xi /˝ ai

modulo the image of a second differential d0;22 .1/. This relation is solved by ai WDOci .P / and determines the image of the vector a WD .a1; : : : ; an/ under H 2.BIZn/!H 2.BIZn/=im.d0;22 .1// DW B uniquely. Note that e ı f .P / is also represented by theimage of the vector a in B . This shows that the right square in (66) commutes.

6.4 T -duality triples and group stacks

6.4.1 Let Rn be the classifying space of T -duality triples introduced in [BRS]. Itcarries a universal T -duality triple tuniv WD ..Euniv;Huniv/; . yEuniv; yHuniv/; uuniv/. Letcuniv; Ocuniv 2 H 2.RnIZn/ be the Chern classes of the bundlesEuniv ! Rn, yEuniv ! Rn.They satisfy the relation cuniv [ Ocuniv D 0. Let Euniv be the extension of sheaves


corresponding toEuniv ! Rn as in 6.3.1. In [BRS] we have shown thatH 3.RnIZ/ Š 0.The diagram (65) now implies that there is a unique Picard stack Puniv 2 QEuniv withOc.Puniv/ D Ocuniv and underlying pair up.Puniv/ Š .Euniv;Huniv/ (see 6.3.5).

6.4.2 Let us fix a Tn-principal bundle E ! B , or the corresponding extension ofsheaves E . Let us furthermore fix a class h 2 F 2H 3.EIZ/. In [BRS] we haveintroduced the set Ext.E; h/ of extensions of .E; h/ to a T -duality triple. The maintheorem about this extension set is [BRS, Theorem 2.24].

Analogously, in the present paper we can consider the set of extensions of .E; h/to a Picard stack P with underlying Tn-bundle E ! B and f .P / D h, wheref W QE ! F 2H 3.EIZ/ is as in (65). In symbols we can write f �1.h/ for this set.

The main goal of the present section is to compare the sets Ext.E; h/ and f �1.h/ �QE . In the following paragraphs we construct maps between these sets.

6.4.3 We fix a Tn-principal bundleE ! B and let E 2 ShAb S=B be the correspondingextensions of sheaves. Let TripleE .B/ denote the set of isomorphism classes of triplest such that c.t/ D c.E/. We first define a map

ˆ W TripleE .B/! QE :

Let t 2 TripleE .B/ be a triple which is classified by a map ft W B ! Rn. Pulling backthe group stackPuniv 2 PIC.S=Rn/we get an elementˆ.t/ WD f �t .Puniv/ 2 PIC.S=B/.If t 2 TripleE .B/, then we have ˆ.t/ 2 QE . We further have

Oc.ˆ.t// D f �t Oc.Puniv/ D f �t Ocuniv D Oc.t/: (67)

6.4.4 In the next few paragraphs we describe a map

‰ W QE ! TripleE .B/;

i.e. a construction of a T -duality triple‰.P / 2 TripleE .B/ starting from a Picard stackP 2 QE .

6.4.5 Consider P 2 PIC.S=B/. We have already constructed one pair .E;H/. Thedual D.P / WD HOMPIC.S=B/.P;BT jB/ is a Picard stack with (see Corollary 5.10)

H 0.D.P // Š D.H�1.P // Š D.T jB/ Š ZjB ;H�1.D.P // Š D.H 0.P // Š D.E/:

In view of the structure (57) of E , the equalities D.TnjB/ Š ZnjB , D.ZjB/ Š T jB , and

Ext1ShAbS=B.T jB ;T jB/ Š 0 we have an exact sequence

0! T jB ! D.E/! ZnjB ! 0: (68)


Using the construction 2.5.12 we can form a quotientD.P /which fits into the sequenceof maps of Picard stacks

BT jB ! D.P /! D.P /;

where H 0.D.P // Š ZjB and H�1.D.P // Š ZnjB . The fibre product

R ��

��

D.P /

��

¹1º �� ZjB

defines a gerbe R! B with band ZnjB .

6.4.6 By Lemma 6.15 there exists a unique isomorphism class yE ! B of a

Tn-bundle whose ZnjB -gerbe of Rn-reductions R yEZn is isomorphic to R. We fix suchan isomorphism and obtain a canonical map of stacks can (see (54)) fitting into thediagram

yH op ��

��

zyH

��

�� D.P /

��

yE can ��

��

��

�� R

yEZn

Š ��

��

R

##��

��

��

��

D.P /

��

B ¹1º �� ZjB .

(69)

The gerbes yH op ! yE and zyH ! R are defined such that the squares become two-cartesian (we omit to write the two-isomorphisms). In this way the Picard stack P 2QE defines the second pair . yE; yH/ of the triple ‰.P / D ..E;H/; . yE; yH/; u/ whoseconstruction has to be completed by providing u.

Lemma 6.18. We have the equality Oc.P / D Oc.‰.P // in H 2.BIZnjB/.

Proof. By the definition in 6.1.12 we have Oc.‰.P // D c. yE/. Furthermore, by (56)

we have c. yE/ D d.R yEZn

jB

/ D d.R/. By Lemma 5.14 we have �.D.P // D D.�.P //,

where � is the characteristic class (46), and

D W QE Š Ext2ShAb S=B.E;T jB/�! Ext2ShAb S=B.D.T jB/;D.E//

is as in Lemma 5.14. The map Oc W QE ! H 2.BIZ/ by its definition fits into the


diagram

QE

a

$$

Oc

��

TnjB!E

��

D

��

Ext2ShAb S=B.TnjB IT jB/

D

��

Ext2ShAb S=B.D.T jB/;D.E//D.E/!D.Tn

jB/�� Ext2ShAb S=B.D.T jB/ID.Tn

jB//

Š��

Ext2ShAb S=B.ZjB ;ZnjB/

Š��

H 2.BIZnjB/.

By construction the class a.P / 2 Ext2ShAb S=B.ZjB ;ZnjB/ classifies the Picard stack

D.E/. This implies that Oc.P / classifies the gerbe R! B , i.e. Oc.P / D c. yE/.6.4.7 It remains to construct the last entry

u W H ˝E�B yE

yH op ! BT jE�B yE (70)

of the triple ‰.t/ D ..E;H/; . yE; yH/; u/, where we use Picture 6.1.11. Note thatH�1.P / Š T jB . Construction 2.5.12 gives rise to a sequence of morphisms of Picardstacks

BT jB ! P ! xP(we could write xP D E).

We have an evaluation ev W P �B D.P / ! BTjB . For a pair .r; s/ 2 Z � Z weconsider the following diagram with two-cartesian squares.

Hr �B zyHs ��

��

P �D.P /

��

ev �� BT jB

Hr ˝Er�B yRszyHs

ur;s

%%

��

w

��

P ˝ xP�D.P/ D.P /

��

Er �B Rs ��

��

xP �D.P /

��¹r; sºjB �� ZjB � ZjB .


By an inspection of the definitions one checks that the natural factorization ur;s existsif r D s. Furthermore one checks that

.w; ur;s/ W Hr ˝Er�BRs zyHs ! .Er �B Rs/ �BT jB

is an isomorphism of gerbes with band T jEr�BRs if and only if r D s D 1. Bothstatements can be checked already when restricting to a point, and therefore becomeclear when considering the argument in the proof of Lemma 6.19.

We define the map (70) by

H ˝E�B yE

yH op

u

��

��

��

H1 ˝E1�B yR1zyH1

��

u1;1�� BT jB

E �B yE.id;can/

�� E1 �B yR1(note that R1 D R, E1 D E and H1 D H ).

Lemma 6.19. The triple ‰.P / D ..E;H/; . yE; yH/; u/ constructed above is a T -duality triple.

Proof. It remains to show that the isomorphism of gerbes u satisfies the condition 6.2,2 in the version of 6.1.11.

By naturality of the construction ‰ in the base B and the fact that condition 6.2, 2can be checked at a single point b 2 B , we can assume without loss of generality thatB is a point. We can further assume that E D Z � Tn and P D BT � Z � Tn. Inthis case

D.P / D D.BT / �D.Z/ �D.Tn/ Š Z �BT �BZn:

We haveH � zyH Š .BT � Tn/ � .BT �BZn/:

The restriction of the evaluation map H � zyH ! H ˝ zyH ! BT is the composition

.BT � Tn/ � .BT �BZn/ Š BT � .BT / � Tn �BZn

id�id�ev��! BT �BT �BT†�! BT :

We are interested in the contribution ev W Tn �BZn ! BTn.It suffices to see that in the case n D 1 we have

ev�.z/ D x ˝ y;where x 2 H 1.T IZ/, y 2 H 1.BZIZ/, and z 2 H 2.BT / Š Z are the canonicalgenerators. In fact this implies via the Künneth formula that ev�.z/ DPn

iD1 xi ˝ yi ,


where xi WD p�i .x/, yi WD q�i .y/ for the projections onto the components pi W Tn !T and qi W BZn Š .BZ/n ! BZ. Finally we use that can�.yi / D xi , wherecan W Tn ! BZn is the canonical map (54) from a second copy of the torus to itsgerbe of Rn-reductions (after identification of this gerbe with BZn).

This finishes the construction of ‰ which started in 6.4.4.

6.4.8 In [BRS, 2.11] we have seen that the groupH 3.BIZ/ acts on Triple.B/ preserv-ing the subsets TripleE .B/ � Triple.B/ for every Tn-bundle E ! B . We will recallthe description of the action in the proof of Lemma 6.20 below. By (65) it also acts onQE .

Lemma 6.20. The map ‰ is H 3.BIZ/-equivariant.

The proof requires some preparations.

6.4.9 Note that we have a canonical isomorphism T Š D.Z/. In order to work withcanonical identifications we are going to use D.Z/ instead of T .

The isomorphisms classes of gerbes with band D.Z/ over a space B 2 S areclassified by

H 2.BID.Z/jB/ Š Ext2ShAb S=B.ZjB ;D.Z/jB/: (71)

The latter group also classifies Picard stacksP with fixed isomorphismsH 0.P / Š ZjBand H�1.P / Š D.ZjB/.

Given P the gerbe G ! B can be reconstructed as a pull-back

G

��

�� P

��¹1ºjB �� ZjB .

If we want to stress the dependence of G on P we will write G.P /.Recall that an object in P.T / as a stack over S consists of a map T ! B and an

object of P.T ! B/. In a similar manner we interpret morphisms.For example, the stack ¹1ºjB is the space B .

6.4.10 Let P 2 ExtPIC.S=B/.ZjB ;D.Z/jB/ be a Picard stack P with a fixed isomor-phismsH 0.P / Š ZjB andH�1.P / Š D.ZjB/. Let � W ExtPIC.S=B/.ZjB ;D.Z/jB/!Ext2ShAb S=B.ZjB ;D.Z/jB/ be the characteristic class.

Lemma 6.21. �.D.P // Š ��.P /.Proof. First of all note that we have canonical isomorphisms

H 0.D.P // Š D.H�1.P / Š D.D.ZjB// Š ZjBand

H�1.D.P // Š D.H 0.P // Š D.ZjB/ Š D.Z/jB :


Therefore we can considerD.P / 2 ExtPIC.S=B/.ZjB ;D.Z/jB/ in a canonical way, and�.P / and �.D.P // belong to the same group.

Note thatd.G.P // D �.P /; d.G.D.P /// D �.D.P //

under the isomorphism (71). It suffices to show that d.G.P // D �d.G.D.P ///.In fact, as in 6.4.7 we have the following factorization of the evaluation map:

G.P /˝BG .D.P //

��

�� P ˝ZjBD.P /

��

��

P �ZjBD.P /

��

�� P �B D.P / ev ��

��

TjB

¹1ºjB

�� ZjB Z

jB

diag�� Z

jB � ZjB .

This represents G.D.P // as the dual gerbe G.P /op of G.P / in the sense of Defini-tion 6.6. The relation d.G.P // D �d.G.D.P /// follows.

6.4.11 We now start the actual proof of Lemma 6.21.In order to see that ‰ is H 3.BIZ/-equivariant we will first describe the action of

H 3.BIZ/ on the sets of isomorphism classes of T -duality triples with fixed underlyingTn-bundles E and yE on the one hand, and on the set of isomorphism classes of Picardstacks QE , on the other.

Consider g 2 H 3.BIZ/. It classifies the isomorphism class of a gerbe G ! B

with band T jB . If t is represented by ..E;H/; . yE; yH/; u/, then g C t is representedby

..E;H ˝ ��G/; . yE; yH ˝ O��G/; u˝ idr�G/;

where the maps �; O�; r are as in (49).Note the isomorphism H 3.BIZ/ Š H 2.BIT / Š Ext2ShAb S=B.ZjB ;T jB/. There-

fore the classg also classifies an isomorphism class of Picard stacks withH 0. zG/ Š ZjBand H�1. zG/ Š T jB . From zG we can derive the gerbe G by the pull-back

G ��

��

zG

��¹1ºjB �� ZjB .

6.4.12 Recall that we consider an extension E 2 ShAb S=B of the form

0! TnjB ! E ! ZjB ! 0:

We consider a Picard stack P 2 ExtPIC.S=B/.E;D.Z/jB/.Let furthermore zG 2 ExtPIC.S=B/.ZjB ;D.Z/jB/. Then we define

P ˝EzG 2 ExtPIC.S=B/.E;D.Z/jB/


by the diagram

BT jB

��

BT jB �B BT jBC��

��

P ˝EzG P �Z

jBzG

��

�� P �B zG

��

ZjBdiag

�� ZjB �B ZjB .

(72)

The right lower square is cartesian, and in the left upper square we take the fibre-wisequotient by the anti-diagonal action of BT jB .

6.4.13 We have D.P / 2 ExtPIC.S=B/.Z;D.E//, where

0! BT jB ! D.E/! D.E/! 0:

We define D.P / to be the quotient of D.P / by BT jB in the sense of 2.5.12 so that

D.P /! D.P / is a gerbe with band T .We define D.P /˝D.P/ D. zG/ by the diagram

BT jB

��

BT jB �B BT jBC��

��

D.P /˝D.P/ D. zG/ D.P / �ZjBD. zG/

��

�� D.P / �B D. zG/

��

ZjBdiag

�� ZjB �B ZjB .

(73)

Again, the right lower square is cartesian, and in the left upper square we take thefibre-wise quotient by the anti-diagonal action of BT jB .

Proposition 6.22. We have an equivalence of Picard stacks

D.P ˝EzG/ Š D.P /˝D.P/ D. zG/:

Proof. The diagram (73) defines D.P /˝D.P/ D. zG/ by forming the pull-back to thediagonal and then taking the quotient of the anti-diagonal BT jB -action in the fibre.One can obtain this diagram by dualizing (72) and interchanging the order of pull-backand quotient.


6.4.14 We can now finish the argument that ‰ is H 3.BIZ/-equivariant. We let‰.P / D ..E;H/; . yE; yH/; u/ and ‰.g C P / D ..E 0;H 0/; . yE 0; yH 0/; u0/. Note thatgCP D P ˝E

zG. An inspection of the construction of the first entry of‰ shows that

E 0 Š E and H 0 Š H ˝E prE ! BG. Proposition 6.22 shows that D.P ˝EzG/ Š

D.P /. This implies that yE 0 Š yE. Furthermore, if we restrict D.P ˝EzG/ along

¹1ºjB ! ZjB we get by Proposition 6.22 and the proof of Lemma 6.21 a diagram ofstacks over B:

yH op ˝ yE Gop

��

�� zyH ˝R

yERGop ��

��

D.P ˝EzG/

��

Š �� D.P /˝D.P/ D. zG/

��

yE can �� RyE

R��

��

D.P ˝EzG/

��

Š �� D.P /

¹1ºjB �� ZjB .

The map u0 is induced by the evaluation .P ˝EzG/ � D.P ˝E

zG/ ! BT . Withthe given identifications using the “duality” between (73) and (72) we see that thisevaluation the induced by the product of the evaluations

.P �D.P // � . zG �D. zG//! BT �BT ! BT :

After restriction to ¹1ºjB we see that u0 D u ˝ v, where v W G ˝ Gop ! BT is thecanonical pairing. This finishes the proof of the equivariance of ‰.

Theorem 6.23. The maps ‰ and ˆ are inverse to each other.

Proof. We first show the assertion under the additional assumption thatH 3.BIZ/ D 0.In this case an elementP 2 QE is determined uniquely by the class Oc.P / 2 H 2.BIZn/.Similarly, a T -duality triple t 2 Triple.B/with c.t/ D c.E/ is uniquely determined by

the class Oc.t/ D c. yE/. Since forP 2 QE we have Oc.ˆı‰.P // (67)D Oc.‰.P // Lemma 6.18DOc.P / and Oc.‰ıˆ.t// Lemma 6.18D Oc.ˆ.t// (67)D Oc.t/ this implies thatˆı‰jQE

D idQEand

‰ ıˆjs�1.E/ D idjs�1.E/, where s W Triple! P is as in 6.1.7. Note thatH 3.RnIZ/ D0. Therefore

‰.ˆ.tuniv// D tuniv; ˆ.‰.Puniv// Š Puniv:

Now consider a general space B 2 S. We first show that ‰ ı ˆ D id. Lett 2 s�1.E/ be classified by the map ft W B ! Rn, i.e. f �t tuniv D t . Then we have‰.ˆ.t// D ‰.f �t Puniv/ D f �t ‰.Puniv/ D f �t tuniv D t , i.e.

‰ ıˆ D id: (74)

We consider the group

�E WD .im.˛/C im.s//=im.˛/ � H 3.BIZ/=im.˛/:


This group is exactly the group ker.��/=C in the notation of [BRS, Theorem 2.24(3)].It follows from (65) that the action of H 3.BIZ/ on QE induces an action of �E onQE which preserves the fibres of f . In [BRS, Theorem 2.24(3)] we have shown that italso acts on TripleE .B/ and preserves the subsets Ext.E; h/.

Let us fix a Picard stack P 2 QE , and let Oc WD Oc.P / and h 2 H 3.EIZ/ be suchthat f .P / 2 Ext.E; h/. From (65) we see that the group �E acts simply transitivelyon the set

AE; Oc WD ¹Q 2 f �1.h/ j Oc.Q/ D Ocº:By [BRS, Theorem 2.24(3)] it also acts simply transitively on the set

BE; Oc WD ¹t 2 Ext.E; h/ j Oc.t/ D Ocº:By Lemma 6.18 we have ‰.AE; Oc/ � BE; Oc . By Lemma 6.20 the map ‰ is �E -equivariant. Hence it must induce a bijection between AE; Oc and BE; Oc . If we let Oc runover all possible choices (solutions of Oc[c.E/ D 0) we see that‰ W QE ! TripleE .B/is a bijection. In view of (74) we now also get ˆ ı‰ D id.

Acknowledgment. We thank Tony Pantev for the crucial suggestion which initiatedthe work on this paper.

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[Brd97] Glen E. Bredon, Sheaf theory, Grad. Texts in Math. 170, Springer-Verlag, New York1997.

[Bre69] Lawrence Breen, Extensions of abelian sheaves and Eilenberg-MacLane algebras, In-vent. Math. 9 (1969/1970), 15–44.

[Bre76] Lawrence Breen, Extensions du groupe additif sur le site parfait, in Algebraic surfaces(Orsay, 1976–78), Lecture Notes in Math. 868, Springer-Verlag, Berlin 1981, 238–262.

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[Bro82] Kenneth S. Brown, Cohomology of groups, Grad. Texts in Math. 87, Springer-Verlag,New York 1982.

[BRS] Ulrich Bunke, Philipp Rumpf, and Thomas Schick, The topology of T -duality forT n-bundles, Rev. Math. Phys. 18 (2006), 1103–1154.

[BS05] Ulrich Bunke and Thomas Schick, On the topology of T -duality, Rev. Math. Phys. 17(2005), 77–112.

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Deformations of gerbes on smooth manifolds

Paul Bressler�, Alexander Gorokhovsky�, Ryszard Nest, and Boris Tsygan�

1 Introduction

In [5] we obtained a classification of formal deformations of a gerbe on a manifold (C1or complex-analytic) in terms of Maurer–Cartan elements of the differential graded Liealgebra (DGLA) of Hochschild cochains twisted by the cohomology class of the gerbe.In the present paper we develop a different approach to the derivation of this classi-fication in the setting of C1 manifolds, based on the differential-geometric approachof [4].

The main result of the present paper is the following theorem which we prove inSection 8.

Theorem 1. Suppose thatX is aC1 manifold and S is an algebroid stack onX whichis a twisted form of OX . Then, there is an equivalence of 2-groupoid valued functorsof commutative Artin C-algebras

DefX .S/ Š MC2.gDR.JX /ŒS�/:

Notations in the statement of Theorem 1 and the rest of the paper are as follows.We consider a paracompact C1-manifold X with the structure sheaf OX of complexvalued smooth functions. Let S be a twisted form of OX , as defined in Section 4.5.Twisted forms of OX are in bijective correspondence with O�X -gerbes and are classifiedup to equivalence by H 2.X IO�/ Š H 3.X IZ/.

One can formulate the formal deformation theory of algebroid stacks ([17], [16])which leads to the 2-groupoid valued functor DefX .S/ of commutative Artin C-alge-bras. We discuss deformations of algebroid stacks in Section 6. It is natural to expectthat the deformation theory of algebroid pre-stacks is “controlled” by a suitably con-structed differential graded Lie algebra (DGLA) well-defined up to isomorphism in thederived category of DGLA. The content of Theorem 1 can be stated as the existence ofsuch a DGLA, namely g.JX /ŒS�, which “controls” the formal deformation theory ofthe algebroid stack S in the following sense.

To a nilpotent DGLA g which satisfies gi D 0 for i < �1 one can associate itsDeligne 2-groupoid which we denote MC2.g/, see [11], [10] and references therein.We review this construction in Section 3. Then Theorem 1 asserts equivalence of the2-groupoids DefX .S/ and MC2.gDR.JX /ŒS�/.

�Supported by the Ellentuck Fund�Partially supported by NSF grant DMS-0400342�Partially supported by NSF grant DMS-0605030

350 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan

The DGLA g.JX /ŒS� is defined as the ŒS �-twist of the DGLA

gDR.JX / WD �.X IDR. xC �

.JX //Œ1�/:

Here, JX is the sheaf of infinite jets of functions on X , considered as a sheaf of topo-logical OX -algebras with the canonical flat connection rcan. The shifted normalizedHochschild complex xC �.JX /Œ1� is understood to comprise locally defined OX -linearcontinuous Hochschild cochains. It is a sheaf of DGLA under the Gerstenhaber bracketand the Hochschild differential ı. The canonical flat connection on JX induces one, alsodenoted rcan, on xC �.JX //Œ1�. The flat connection rcan commutes with the differentialı and acts by derivations of the Gerstenhaber bracket. Therefore, the de Rham complexDR. xC �.JX //Œ1�/ WD ��

X ˝ xC �.JX //Œ1�/ equipped with the differential rcan C ı andthe Lie bracket induced by the Gerstenhaber bracket is a sheaf of DGLA on X givingrise to the DGLA g.JX / of global sections.

The sheaf of abelian Lie algebras JX=OX acts by derivations of degree �1 on thegraded Lie algebra xC �.JX /Œ1� via the adjoint action. Moreover, this action commuteswith the Hochschild differential. Therefore, the (abelian) graded Lie algebra ��

X ˝JX=OX acts by derivations on the graded Lie algebra ��

X ˝ xC �.JX //Œ1�. We denotethe action of the form ! 2 ��

X ˝ JX=OX by �! . Consider now the subsheaf of closedforms .��

X˝JX=OX /cl which is by definition the kernel ofrcan. .�kX˝JX=OX /

cl actsby derivations of degree k � 1 and this action commutes with the differential rcanC ı.Therefore, for ! 2 �.X I .�2 ˝ JX=OX /

cl/ one can define the !-twist g.JX /! asthe DGLA with the same underlying graded Lie algebra structure as g.JX / and thedifferential given byrcanCıC �! . The isomorphism class of this DGLA depends onlyon the cohomology class of ! in H 2.�.X I��

X ˝ JX=OX /;rcan/.More precisely, for ˇ 2 �.X I�1X ˝ JX=OX / the DGLA gDR.JX /! and

gDR.JX /!Crcanˇ are canonically isomorphic with the isomorphism depending onlyon the equivalence class ˇ C Im.rcan/.

As we remarked before a twisted form S of OX is determined up to equivalence by its

class in H 2.X IO�/. The composition O� ! O�=C�log�! O=C

j1

��! DR.J=O/ in-duces the mapH 2.X IO�/! H 2.X IDR.J=O// Š H 2.�.X I��

X˝JX=OX /;rcan/.We denote by ŒS � 2 H 2.�.X I��

X ˝ JX=OX /;rcan/ the image of the class of S . Bythe remarks above we have the well-defined up to a canonical isomorphism DGLAgDR.JX /ŒS�.

The rest of this paper is organized as follows. In Section 2 we review some prelim-inary facts. In Section 3 we review the construction of Deligne 2-groupoid, its relationwith the deformation theory and its cosimplicial analogues. In Section 4 we review thenotion of algebroid stacks. Next we define matrix algebras associated with a descentdatum in Section 5. In Section 6 we define the deformations of algebroid stacks andrelate them to the cosimplicial DGLA of Hochschild cochains on matrix algebras. InSection 7 we establish quasiisomorphism of the DGLA controlling the deformationsof twisted forms of OX with a simpler cosimplicial DGLA. Finally, the proof the mainresult of this paper, Theorem 1, is given in Section 8

Deformations of gerbes on smooth manifolds 351

2 Preliminaries

2.1 Simplicial notions

2.1.1 The category of simplices. For n D 0; 1; 2; : : : we denote by Œn� the categorywith objects 0; : : : ; n freely generated by the graph

0! 1! � � � ! n:

For 0 � p � q � n we denote by .pq/ the unique morphism p ! q.We denote by � the full subcategory of Cat with objects the categories Œn� for

n D 0; 1; 2; : : :.For 0 � i � nC 1 we denote by @i D @ni W Œn�! ŒnC 1� the i th face map, i.e. the

unique map whose image does not contain the object i 2 ŒnC 1�.For 0 � i � n � 1 we denote by si D sni W Œn� ! Œn � 1� the i th degeneracy map,

i.e. the unique surjective map such that si .i/ D si .i C 1/.

2.1.2 Simplicial and cosimplicial objects. Suppose that C is a category. By defi-nition, a simplicial object in C (respectively, a cosimplicial object in C ) is a functor�op ! C (respectively, a functor � ! C ). Morphisms of (co)simplicial objects arenatural transformations of functors.

For a simplicial (respectively, cosimplicial) object F we denote the object F.Œn�/ 2C by Fn (respectively, F n).

2.2 Cosimplicial vector spaces. Let V � be a cosimplicial vector space. We denoteby C �.V / the associated complex with component C n.V / D V n and the differential@n W C n.V / ! C nC1.V / defined by @n D P

i .�1/i@ni , where @ni is the map inducedby the i th face map Œn�! ŒnC 1�. We denote cohomology of this complex byH �.V /.

The complex C �.V / contains a normalized subcomplex xC �.V /. Here xC n.V / DfV 2 V n j sni v D 0g, where sni W Œn�! Œn � 1� is the i th degeneracy map. Recall thatthe inclusion xC �.V /! C �.V / is a quasiisomorphism.

Starting from a cosimplicial vector space V � one can construct a new cosimplicialvector space yV � as follows. For every � W Œn� ! � set yV � D V �.n/. Suppose givenanother simplex � W Œm� ! � and morphism W Œm� ! Œn� such that � D � ı , i.e., is a morphism of simplices �! �. The morphism .0n/ factors uniquely into 0!.0/ ! .m/ ! n, which, under �, gives the factorization of �.0n/ W �.0/ ! �.n/

(in �) into

�.0/f��! �.0/

g��! �.m/h�! �.n/; (2.1)

where g D �.0m/. The map �.m/! �.n/ induces the map

� W yV � ! yV �: (2.2)

Set now yV n D QŒn�

��!�

yV �. The maps (2.2) endow yV � with the structure of a cosim-

plicial vector space. We then have the following well-known result:


Lemma 2.1. H �.V / Š H �. yV /:Proof. We construct morphisms of complexes inducing the isomorphisms in cohomol-ogy. We will use the following notations. If f 2 yV n and � W Œn�! � we will denoteby f .�/ 2 yV � its component in yV �. For � W Œn�! � we denote by �jŒjl� W Œl � j �!� its truncation: �jŒjl�.i/ D �.i C j /, �jŒjl�.i; k/ D �..i C j / .k C j //. For�1 W Œn1� ! � and �2 W Œn2� ! � with �1.n1/ D �2.0/ define their concatenationƒ D �1 � �2 W Œn1 C n2�! � by the following formulas.

ƒ.i/ D´�1.i/ if i � n1;�2.i � n1/ if i � n1;

ƒ.ik/ D

8<:�1.ik/ if i; k � n1;�2..i � n1/ .k � n1// if i; k � n1;�2.0 .k � n1// ı �.in1/ if i � n1 � k:

This operation is associative. Finally we will identify in our notations � W Œ1�! �withthe morphism �.01/ in �.

The morphism C �.V / ! C �. yV / is constructed as follows. Let � W Œn� ! � be asimplex in � and define �k by �.k/ D Œ�k�, k D 0; 1; : : : ; n. Let ‡.�/ W Œn� ! �.n/

be a morphism in � defined by

.‡.�//.k/ D �.kn/.�k/: (2.3)

Then define the map � W V � ! yV � by the formula

.�.v//.�/ D ‡.�/�v for v 2 V n:This is a map of cosimplicial vector spaces, and therefore it induces a morphism ofcomplexes.

The morphism W C �. yV /! C �.V / is defined by the formula

.f / D .�1/n.nC1/2

X0�ik�kC1

.�1/i0C��Cin�1f .@0i0 � @1i1 � � � � � @n�1in�1/ for f 2 yV n

when n > 0, and .f / is V 0 component of f if n D 0.The morphism � ı is homotopic to Id with the homotopy h W C �. yV /! C ��1. yV /

given by the formula

hf .�/ Dn�1XjD0

X0�ik�kC1

.�1/i0C��Cij�1f .@0i0 � � � � � @j�1ij�1� ‡.�jŒ0j �/ � �jŒj .n�1/�/

for f 2 yV n when n > 0, and h.f / D 0 if n D 0.The composition ı � W C �.V / ! C �.V / preserves the normalized subcomplex

xC �.V / and acts as the identity on it. Therefore ı � induces the identity map oncohomology. It follows that and � are quasiisomorphisms inverse to each other.


2.3 Covers. A cover (open cover) of a space X is a collection U of open subsets ofX such that

SU2U U D X .

2.3.1 The nerve of a cover. LetN0U DÙ2U U . There is a canonical augmentation

map

�0 W N0UÙ2U.U ,!X/��! X:

LetNpU D N0U �X � � � �X N0U

be the .p C 1/-fold fiber product.The assignment NU W � 3 Œp� 7! NpU extends to a simplicial space called the

nerve of the cover U. The effect of the face map @ni (respectively, the degeneracymap sni ) will be denoted by d i D d in (respectively, &i D &ni ) and is given by theprojection along the i th factor (respectively, the diagonal embedding on the i th factor).Therefore for every morphism f W Œp�! Œq� in� we have a morphism NqU! NpU

which we denote by f �. We will denote by f� the operation .f �/� of pull-back alongf �; if F is a sheaf on NpU then f�F is a sheaf on NqU.

For 0 � i � n C 1 we denote by prni W NnU ! N0U the projection onto the i th

factor. For 0 � j � m, 0 � ij � n the map pri0 � � � � � prim W NnU ! .N0U/m

can be factored uniquely as a composition of a map NnU! NmU and the canonicalimbedding NmU! .N0U/

m. We denote this map NnU! NmU by prni0:::im .The augmentation map �0 extends to a morphism � W NU! X where the latter is

regarded as a constant simplicial space. Its component of degree n �n W NnU! X isgiven by the formula �n D �0 ı prni . Here 0 � i � nC 1 is arbitrary.

2.3.2 Cech complex. Let F be a sheaf of abelian groups onX . One defines a cosim-plicial group LC �.U;F / D �.N�UI ��F /, with the cosimplicial structure inducedby the simplicial structure of NU. The associated complex is the Cech complex ofthe cover U with coefficients in F . The differential L@ in this complex is given byP.�1/i .d i /�.

2.3.3 Refinement. Suppose that U and V are two covers ofX . A morphism of covers� W U! V is a map of sets � W U! V with the property U � �.U / for all U 2 U.

A morphism � W U ! V induces the map N� W NU ! NV of simplicial spaceswhich commutes with respective augmentations to X . The map N0� is determined bythe commutativity of

U ��

��

N0U

N0�

��

�.U / �� N0V .

It is clear that the mapN0� commutes with the respective augmentations (i.e. is a map ofspaces over X ) and, consequently induces maps Nn� D .N0�/�XnC1 which commutewith all structure maps.


2.3.4 The category of covers. Let Cov.X/0 denote the set of open covers of X . ForU;V 2 Cov.X/0 we denote by Cov.X/1.U;V/ the set of morphisms U ! V . LetCov.X/ denote the category with objects Cov.X/0 and morphisms Cov.X/1. Theconstruction of 2.3.1 is a functor

N W Cov.X/! Top�op=X:

3 Deligne 2-groupoid and its cosimplicial analogues

We begin this section by recalling the definition of Deligne 2-groupoid and its relationwith the deformation theory. We then describe the cosimplicial analogues of Deligne2-groupoids and establish some of their properties.

3.1 Deligne 2-groupoid. In this subsection we review the construction of Deligne2-groupoid of a nilpotent differential graded algebra (DGLA). We follow [11], [10] andreferences therein.

Suppose that g is a nilpotent DGLA such that gi D 0 for i < �1.A Maurer–Cartan element of g is an element 2 g1 satisfying

d C 1

2Œ ; � D 0: (3.1)

We denote by MC2.g/0 the set of Maurer–Cartan elements of g.The unipotent group exp g0 acts on the set of Maurer–Cartan elements of g by the

gauge equivalences. This action is given by the formula

.expX/ � D �1XiD0

.adX/i

.i C 1/Š .dX C Œ ; X�/

If expX is a gauge equivalence between two Maurer–Cartan elements 1 and 2 D.expX/ � 1 then

d C ad 2 D Ad expX .d C ad 1/: (3.2)

We denote by MC2.g/1. 1; 2/ the set of gauge equivalences between 1, 2. Thecomposition

MC2.g/1. 2; 3/ �MC2.g/1. 1; 2/! MC2.g/1. 1; 3/

is given by the product in the group exp g0.If 2 MC2.g/0 we can define a Lie bracket Œ�; �� on g�1 by

Œa; b� D Œa; db C Œ ; b��: (3.3)

With this bracket g�1 becomes a nilpotent Lie algebra. We denote by exp g�1the corresponding unipotent group, and by exp the corresponding exponential mapg�1 ! exp g�1. If 1, 2 are two Maurer–Cartan elements, then the group exp g�1


acts on MC2.g/1. 1; 2/. Let exp t 2 exp g�1 and let expX 2 MC2.g/1. 1; 2/.Then

.exp t / � .expX/ D exp.dt C Œ ; t �/ expX 2 exp g0:

Such an element exp t is called a 2-morphism between expX and .exp t / �.expX/. Wedenote by MC2.g/2.expX; expY / the set of 2-morphisms between expX and expY .This set is endowed with a vertical composition given by the product in the groupexp g�1.

Let 1; 2; 3 2 MC2.g/0. Let expX12; expY12 2 MC2.g/1. 1; 2/ and expX23,expY23 2 MC2.g/1. 2; 3/. Then one defines the horizontal composition

˝W MC2.g/2.expX23; expY23/ �MC2.g/2.expX12; expY12/

! MC2.g/2.expX23 expX12; expX23 expY12/

as follows. Let

exp2 t12 2 MC2.g/2.expX12; expY12/; exp3 t23 2 MC2.g/2.expX23; expY23/:

Then

exp3 t23 ˝ exp2 t12 D exp3 t23 exp3.eadX23.t12//

To summarize, the data described above forms a 2-groupoid which we denote byMC2.g/ as follows:

1. the set of objects is MC2.g/0,

2. the groupoid of morphisms MC2.g/. 1; 2/, i 2 MC2.g/0 consists of

• objects, i.e., 1-morphisms in MC2.g/ are given by MC2.g/1. 1; 2/ – thegauge transformations between 1 and 2,

• morphisms between expX , expY 2 MC2.g/1. 1; 2/ which are given byMC2.g/2.expX; expY /.

A morphism of nilpotent DGLA W g ! h induces a functor W MC2.g/ !MC2.g/.

We have the following important result ([12], [11] and references therein).

Theorem 3.1. Suppose that W g! h is a quasi-isomorphism of DGLA and let m be anilpotent commutative ring. Then the induced map W MC2.g˝m/! MC2.h˝m/is an equivalence of 2-groupoids.

3.2 Deformations and Deligne 2-groupoid. Let k be an algebraically closed field ofcharacteristic zero.


3.2.1 Hochschild cochains. Suppose that A is a k-vector space. The k-vector spaceC n.A/ of Hochschild cochains of degree n � 0 is defined by

C n.A/ WD Homk.A˝n; A/:

The graded vector space g.A/ WD C �.A/Œ1� has a canonical structure of a graded Liealgebra under the Gerstenhaber bracket denoted by Œ ; � below. Namely, C �.A/Œ1� iscanonically isomorphic to the (graded) Lie algebra of derivations of the free associativeco-algebra generated by AŒ1�.

Suppose in addition that A is equipped with a bilinear operation � W A˝ A ! A,i.e. � 2 C 2.A/ D g1.A/. The condition Œ�; �� D 0 is equivalent to the associativityof �.

Suppose that A is an associative k-algebra with the product �. For a 2 g.A/ letı.a/ D Œ�; a�. Thus, ı is a derivation of the graded Lie algebra g.A/. The associativityof � implies that ı2 D 0, i.e. ı defines a differential on g.A/ called the Hochschilddifferential.

For a unital algebra the subspace of normalized cochains xC n.A/ C n.A/ is definedby

xC n.A/ WD Homk..A=k � 1/˝n; A/:The subspace xC �.A/Œ1� is closed under the Gerstenhaber bracket and the action of theHochschild differential and the inclusion xC �.A/Œ1� ,! C �.A/Œ1� is a quasi-isomorphismof DGLA.

Suppose in addition that R is a commutative Artin k-algebra with the nilpotentmaximal ideal mR The DGLA g.A/˝k mR is nilpotent and satisfies gi .A/˝k mR D0 for i < �1. Therefore, the Deligne 2-groupoid MC2.g.A/ ˝k mR/ is defined.Moreover, it is clear that the assignmentR 7! MC2.g.A/˝k mR/ extends to a functoron the category of commutative Artin algebras.

3.2.2 Star products. Suppose that A is an associative unital k-algebra. Letm denotethe product on A.

LetR be a commutative Artin k-algebra with maximal ideal mR. There is a canon-ical isomorphism R=mR Š k.

An (R-)star product on A is an associativeR-bilinear product on A˝k R such thatthe canonical isomorphism of k-vector spaces .A˝k R/˝R k Š A is an isomorphismof algebras. Thus, a star product is an R-deformation of A.

The 2-category of R-star products on A, denoted Def.A/.R/, is defined as thesubcategory of the 2-category Alg2R of R-algebras (see 4.1.1) with

• Objects: R-star products on A,

• 1-morphisms W m1 ! m2 between the star products �i thoseR-algebra homo-morphisms W .A˝kR;m1/! .A˝kR;m2/which reduce to the identity mapmodulo mR, i.e. ˝R k D IdA,


• 2-morphisms b W ! , where ; W m1 ! m2 are two 1-morphisms, areelements b 2 1C A˝k mR A˝k R such that m2..a/; b/ D m2.b; .a//

for all a 2 A˝k R.

It follows easily from the above definition and the nilpotency of mR that Def.A/.R/is a 2-groupoid.

Note that Def.A/.R/ is non-empty: it contains the trivial deformation, i.e. the starproduct, still denoted m, which is the R-bilinear extension of the product on A.

It is clear that the assignmentR 7! Def.A/.R/ extends to a functor on the categoryof commutative Artin k-algebras.

3.2.3 Star products and the Deligne 2-groupoid. We continue in notations intro-duced above. In particular, we are considering an associative unital k-algebra A. Theproduct m 2 C 2.A/ determines a cochain, still denoted m 2 g1.A/˝k R, hence theHochschild differential ı D Œm; � in g.A/˝kR for any commutativeArtin k-algebraR.

Suppose thatm0 is anR-star product onA. Since �.m0/ WD m0�m D 0 mod mR

we have �.m0/ 2 g1.A/˝k mR. Moreover, the associativity ofm0 implies that �.m0/satisfies the Maurer–Cartan equation, i.e. �.m0/ 2 MC2.g.A/˝k mR/0.

It is easy to see that the assignment m0 7! �.m0/ extends to a functor

Def.A/.R/! MC2.g.A/˝k mR/: (3.4)

The following proposition is well known (cf. [9], [11], [10]).

Proposition 3.2. The functor (3.4) is an isomorphism of 2-groupoids.

3.2.4 Star products on sheaves of algebras. The above considerations generalize tosheaves of algebras in a straightforward way.

Suppose that A is a sheaf of k-algebras on a spaceX . Letm W A˝kA! A denotethe product.

AnR-star product on A is a structure of a sheaf of an associative algebras on A˝kRwhich reduces to � modulo the maximal ideal mR. The 2-category (groupoid) of Rstar products on A, denoted Def.A/.R/ is defined just as in the case of algebras; weleave the details to the reader.

The sheaf of Hochschild cochains of degree n is defined by

C n.A/ WD Hom.A˝n;A/:

We have the sheaf of DGLA g.A/ WD C �.A/Œ1�, and hence the nilpotent DGLA�.X Ig.A/˝k mR/ for every commutative Artin k-algebra R concentrated in degrees� �1. Therefore, the 2-groupoid MC2.�.X Ig.A/˝k mR/ is defined.

The canonical functor Def.A/.R/! MC2.�.X Ig.A/˝k mR/ defined just as inthe case of algebras is an isomorphism of 2-groupoids.


3.3 G -stacks. Suppose that G W Œn�! Gn is a cosimplicial DGLA. We assume thateach Gn is a nilpotent DGLA. We denote its component of degree i by Gn;i and assumethat Gn;i D 0 for i < �1.

Definition 3.3. A G-stack is a triple D . 0; 1; 2/, where

• 0 2 MC2.G0/0,

• 1 2 MC2.G1/1.@00 0; @01

0/,

satisfying the conditions10

1 D Id;

• 2 2 MC2.G2/2.@12.

1/ ı @10. 1/; @11. 1//satisfying the conditions

@22 2 ı .Id˝ @20 2/ D @21 2 ı .@23 2 ˝ Id/; and s20 2 D s21 2 D Id: (3.5)

Let Stack.G/0 denote the set of G-stacks.

Definition 3.4. For 1; 2 2 Stack.G/0 a 1-morphism Ç W 1 ! 2 is a pair Ç D.Ç1; Ç2/, where Ç1 2 MC2.G0/1.

01 ;

02 /, Ç2 2 MC2.G1/2.

12 ı @00.Ç1/; @01.Ç1/ ı 11 /,

satisfying

.Id˝ 21 / ı .@12Ç2 ˝ Id/ ı .Id˝ @10Ç2/ D @11Ç2 ı . 22 ˝ Id/;

s10Ç2 D Id:(3.6)

Let Stack.G/1. 1; 2/ denote the set of 1-morphisms 1 ! 2.Composition of 1-morphisms Ç W 1 ! 2 and È W 2 ! 3 is given by .È1 ı Ç1;

.Ç2 ˝ Id/ ı .Id˝ È2//.

Definition 3.5. For Ç1; Ç2 2 Stack.G/1. 1; 2/ a 2-morphism W Ç1 ! Ç2 is a 2-mor-phism 2 MC2.G0/2.Ç11; Ç12/ which satisfies

Ç22 ı .Id˝ @00/ D .@01 ˝ Id/ ı Ç21: (3.7)

Let Stack.G/2.Ç1; Ç2/ denote the set of 2-morphisms.Compositions of 2-morphisms are given by the compositions in MC2.G0/2.For 1; 2 2 Stack.G/0, we have the groupoid Stack.G/. 1; 2/ with the set of

objects Stack.G/1. 1; 2/ and the set of morphisms Stack.G/2.Ç1; Ç2/ under verticalcomposition.

Every morphism � of cosimplicial DGLA induces in an obvious manner a functor�� W Stack.G ˝m/! Stack.H˝m/

We have the following cosimplicial analogue of Theorem 3.1:

Theorem 3.6. Suppose that � W G ! H is a quasi-isomorphism of cosimplicial DGLAand m is a commutative nilpotent ring. Then the induced map �� W Stack.G ˝m/!Stack.H˝m/ is an equivalence.


Proof. The proof can be obtained by applying Theorem 3.1 repeatedly.Let 1, 2 2 Stack.G ˝ m/0, and let Ç1, Ç2 be two 1-morphisms between 1

and 2. Note that �� W MC2.G0 ˝ m/2.Ç11; Ç12/ ! MC2.H0 ˝ m/2.��Ç11; ��Ç12/ isa bijection by Theorem 3.1. Injectivity of the map �� W Stack.G ˝ m/2.Ç1; Ç2/ !Stack.H˝m/2.��Ç1; ��Ç2/ follows immediately.

For the surjectivity, note that an element of Stack.H˝m/2.��Ç1; ��Ç2/ is necessarilygiven by �� for some 2 MC2.G0˝m/.Ç11; Ç12/2 and the following identity is satisfiedin MC2.H1 ˝m/.��. 12 ı @00Ç11/; ��.@01Ç12 ı 11 //2:

��.Ç22 ı .@10 ˝ Id// D ��..Id˝ @11/ ı Ç21/:

Since �� W MC2.G1 ˝m/. 12 ı @00Ç11; @01Ç12 ı 11 /2 ! MC2.H1 ˝m/.��. 12 ı @00Ç11/;��.@01Ç12ı 11 //2 is bijective, and in particular injective, Ç22ı.@10˝Id/ D .Id˝@11/ıÇ21,and defines an element in Stack.G ˝m/2.Ç1; Ç2/.

Next, let 1, 2 2 Stack.G ˝ m/0, and let Ç be a 1-morphisms between �� 1and �� 2. We show that there exists È 2 Stack.G ˝ m/1. 1; 2/ such that ��È isisomorphic to Ç. Indeed, by Theorem 3.1, there exists È1 2 MC2.G0 ˝m/1. 1; 2/such that MC2.H0˝m/2.��È1; Ç/ ¤ ;. Let 2 MC2.H0˝m/2.��È1; Ç/. Define 2MC2.H1˝m/2.��. 12 ı@00È1/; ��.@01È1 ı 11 // by D .@01˝ Id/�1 ıÇ2 ı.Id˝@00/.It is easy to verify that the following identities holds:

.Id˝ �� 21 / ı .@12 ˝ Id/ ı .Id˝ @10 / D @11 ı .�� 22 ˝ Id/;

s10 D Id:

By bijectivity of �� on MC2 there exists a unique È2 2 MC2.H1˝m/2. 12 ı@00È1; @01È1ı

11 / such that ��È2 D . Moreover, as before, injectivity of �� implies that theconditions (3.6) are satisfied. Therefore È D .È1;È2/ defines a 1-morphism 1 ! 2and is a 2-morphism ��È! Ç.

Now, let 2 Stack.H˝m/0. We construct � 2 Stack.G˝m/0 in such a way thatStack.H˝m/1.��; / ¤ ¿. By Theorem 3.1 there exists �0 2 MC2.G0˝m/0 suchthat MC2.H0 ˝m/1.��0; 0/ ¤ ¿. Let Ç1 2 MC2.H0 ˝m/1.��; /. ApplyingTheorem 3.1 again we obtain that there exists � 2 MC2.G1 ˝m/1.@

00�0; @01�

0/ suchthat there exists 2 MC2.H1 ˝ m/2.

1 ı @00Ç1; @01Ç1 ı ��/. Then s10 is then a2-morphism Ç1 ! Ç1 ı .s10�/, which induces a 2-morphism W .s10�/�1 ! Id.

As a next step set �1 D � ı [email protected]�//�1 2 MC2.G1 ˝m/1.@00�0; @01�

0/, Ç2 D˝.@00 /�1 2 MC2.H1˝m/2.

1ı@00Ç1; @01Ç1ı��1/. It is easy to see that s10�1 D Id,

s10Ç2 D Id.We then conclude that there exists a unique �2 such that

.Id˝ ��2/ ı .@12Ç2 ˝ Id/ ı .Id˝ @10Ç2/ D @11Ç2 ı . 2 ˝ Id/:

Such a �2 necessarily satisfies the conditions

@22�2 ı .Id˝ @20�2/ D @21�2 ı .@23�2 ˝ Id/;

s20�2 D s21�2 D Id:


Therefore � D .�0; �1; �2/ 2 Stack.G ˝m/, and Ç D .Ç1; Ç2/ defines a 1-morphism��! .

3.4 Acyclicity and strictness

Definition 3.7. A G-stack . 0; 1; 2/ is called strict if @00 0 D @01

0, 1 D Id and 2 D Id.

Let Stackstr.G/0 denote the subset of strict G-stacks.

Lemma 3.8. Stackstr.G/0 D MC2.ker.G0 � G1//0.

Definition 3.9. For elements 1; 2 in Stackstr.G/0, a 1-morphism Ç D .Ç1; Ç2/ 2Stack.G/1. 1; 2/ is called strict if @00.Ç1/ D @01.Ç1/ and Ç2 D Id.

For 1; 2 2 Stackstr.G/0 we denote by Stackstr.G/1. 1; 2/ the subset of strictmorphisms.

Lemma 3.10. For 1; 2 2 Stackstr.G/0 one has

Stackstr.G/1. 1; 2/ D MC2.ker.G0 � G1//1:

For 1; 2 2 Stackstr.G/0 let Stackstr.G/. 1; 2/ denote the full subcategory ofStack.G/. 1; 2/ with objects Stackstr.G/1. 1; 2/.

Thus, we have the 2-groupoids Stack.G/ and Stackstr.G/ and an embedding of thelatter into the former which is fully faithful on the respective groupoids of 1-morphisms.

Lemma 3.11. Stackstr.G/ D MC2.ker.G0 � G1//.

Suppose that G is a cosimplicial DGLA. For each n and i we have the vectorspace Gn;i , namely the degree i component of Gn. The assignment n 7! Gn;i is acosimplicial vector space G�;i .

We will consider the following acyclicity condition on the cosimplicial DGLA G:

Hp.G�;i / D 0 for all i 2 Z and for p ¤ 0. (3.8)

Theorem 3.12. Suppose that G is a cosimplicial DGLA which satisfies the condition(3.8), and m a commutative nilpotent ring. Then, the functor � W Stackstr.G ˝m/ !Stack.G ˝m// is an equivalence.

Proof. As we already noted before, it is immediate from the definitions that if 1, 2 2 Stackstr.G ˝ m/0, and Ç1; Ç2 2 Stackstr.G ˝ m/1. 1; 2/, then the map-ping � W Stackstr.G ˝m/2.Ç1; Ç2/! Stackstr.G ˝m/2.Ç1; Ç2/ is a bijection.

Let now 1; 2 2 Stackstr.G ˝m/0, Ç D .Ç1; Ç2/ 2 Stack.G ˝m/1. 1; 2/. Weshow that there exists È 2 Stackstr.G/1. 1; 2/ and a 2-morphism W Ç ! È. LetÇ2 D exp@0

002g, where g 2 .G1 ˝ m/. Then @10g � @11g C @12g D 0 mod m2.

As a consequence of the acyclicity condition there exists a 2 .G0 ˝ m/ such that@00a� @01a D g mod m2. Set 1 D exp0

2a. Define then È1 D .1 � Ç1; .@011˝ Id/ı


Ç2 ı .Id ˝ @001/�1/ 2 Stack.G ˝m/1. 1; 2/. Note that 1 defines a 2-morphismÇ! È1. Note also that È21 2 exp@0

002.G˝m2/. Proceeding inductively one constructs

a sequence Èk 2 Stack.G ˝m/1. 1; 2/ such that È2k2 exp.G ˝mkC1/ and 2-mor-

phisms k W Ç! Èk , kC1 D k mod mk . Since m is nilpotent, for k large enoughwe have Èk 2 Stackstr.G/1. 1; 2/.

Assume now that 2 Stack.G ˝m/0. We will construct � 2 Stackstr.G ˝m/0and a 1-morphism Ç W ! �.

We begin by constructing� 2 Stack.G˝m/0 such that�2 D Id and a 1-morphismÈ W ! �. We have: 2 D exp@1

1@010 c, c 2 .G2˝m/. In view of the equations (3.5)

c satisfies the identities

@20c � @21c C @22c � @23c D 0 mod m2; and s20c D s21c D 0: (3.9)

By the acyclicity of the normalized complex we can find b 2 G1 ˝m such that@10b � @11b C @12b D c mod m2, s10b D 0. Let 1 D exp@0

10.�b/. Define then

�1 D .�01; �11; �21/ where �01 D 0, �11 D 1 � 1, and �21 is such that

.Id˝ 2/ ı .@121 ˝ Id/ ı .Id˝ @101/ D @111 ı .�21 ˝ Id/:

Note that �21 D Id mod m2 and .Id; 1/ is a 1-morphism ! �1. As beforewe can construct a sequence �k such that �2

kD Id mod mkC1, and 1-morphisms

.Id; k/ W ! �k , kC1 D k mod mk . As before we conclude that this gives thedesired construction of �. The rest of the proof, i.e. the construction of � is completelyanalogous.

Corollary 3.13. Suppose that G is a cosimplicial DGLA which satisfies the condition(3.8). Then there is a canonical equivalence:

Stack.G ˝m/ Š MC2.ker.G0 � G1/˝m/:

Proof. Combine Lemma 3.11 with Theorem 3.12.

4 Algebroid stacks

In this section we review the notions of algebroid stack and twisted form. We alsodefine the notion of descent datum and relate it with algebroid stacks.

4.1 Algebroids and algebroid stacks

4.1.1 Algebroids. For a category C we denote by iC the subcategory of isomorphismsin C ; equivalently, iC is the maximal subgroupoid in C .

Suppose that R is a commutative k-algebra.

Definition 4.1. An R-algebroid is a nonempty R-linear category C such that thegroupoid iC is connected


Let AlgdR denote the 2-category ofR-algebroids (full 2-subcategory of the 2-cate-gory of R-linear categories).

Suppose that A is an R-algebra. The R-linear category with one object and mor-phisms A is an R-algebroid denoted AC.

Suppose that C is an R-algebroid and L is an object of C . Let A D EndC .L/. Thefunctor AC ! C which sends the unique object of AC to L is an equivalence.

Let Alg2R denote the 2-category with

• objects: R-algebras,

• 1-morphisms: homomorphism of R-algebras,

• 2-morphisms ! , where ; W A ! B are two 1-morphisms: elementsb 2 B such that .a/ � b D b � .a/ for all a 2 A.

It is clear that the 1- and the 2- morphisms in Alg2R as defined above induce 1- and2-morphisms of the corresponding algebroids under the assignment A 7! AC. Thestructure of a 2-category on Alg2R (i.e. composition of 1- and 2-morphisms) is deter-mined by the requirement that the assignment A 7! AC extends to an embedding.�/C W Alg2R ! AlgdR.

Suppose that R ! S is a morphism of commutative k-algebras. The assignmentA! A˝R S extends to a functor .�/˝R S W Alg2R ! Alg2S .

4.1.2 Algebroid stacks. We refer the reader to [1] and [20] for basic definitions. Wewill use the notion of fibered category interchangeably with that of a pseudo-functor.A prestack C on a space X is a category fibered over the category of open subsetsof X , equivalently, a pseudo-functor U 7! C.U /, satisfying the following additionalrequirement. For an open subset U of X and two objects A;B 2 C.U / we have thepresheaf HomC .A;B/ on U defined by U V 7! HomC.V /.AjV ; BjB/. The fiberedcategory C is a prestack if for any U , A;B 2 C.U /, the presheaf HomC .A;B/ is asheaf. A prestack is a stack if, in addition, it satisfies the condition of effective descentfor objects. For a prestack C we denote the associated stack by zC .

Definition 4.2. A stack in R-linear categories C on X is an R-algebroid stack if it islocally nonempty and locally connected, i.e. satisfies

1. any point x 2 X has a neighborhood U such that C.U / is nonempty;

2. for any U � X , x 2 U , A;B 2 C.U / there exits a neighborhood V � U of xand an isomorphism AjV Š BjV .

Remark 4.3. Equivalently, the stack associated to the substack of isomorphisms �iC isa gerbe.

Example 4.4. Suppose that A is a sheaf of R-algebras on X . The assignment X U 7! A.U /C extends in an obvious way to a prestack in R-algebroids denoted AC.


The associated stack �AC is canonically equivalent to the stack of locally free Aop-modules of rank one. The canonical morphism AC ! �AC sends the unique (locallydefined) object of AC to the free module of rank one.

1-morphisms and 2-morphisms ofR-algebroid stacks are those of stacks inR-linearcategories. We denote the 2-category of R-algebroid stacks by AlgStackR.X/.

4.2 Descent data

4.2.1 Convolution data

Definition 4.5. An R-linear convolution datum is a triple .U;A01;A012/ consistingof

• a cover U 2 Cov.X/,

• a sheaf A01 of R-modules A01 on N1U,

• a morphism

A012 W .pr201/�A01 ˝R .pr212/

�A01 ! .pr202/�A01 (4.1)

of R-modules

subject to the associativity condition expressed by the commutativity of the diagram

.pr301/�A01 ˝R .pr312/

�A01 ˝R .pr323/�A01

Id˝.pr3123/�.A012

��

.pr3012/�.A012/˝Id

�� .pr302/�A01 ˝R .pr323/

�A01

.pr3023/�.A012/

��

.pr301/�A01 ˝R .pr313/

�A01

.pr3013/�.A012/

�� .pr303/�A01.

For a convolution datum .U;A01;A012/we denote by A the pair .A01;A012/ andabbreviate the convolution datum by .U;A/.

For a convolution datum .U;A/ let

• A WD .pr000/�A01; A is a sheaf of R-modules on N0U,

• Api WD .prpi /

�A; thus for every p we get sheaves Api , 0 � i � p on NpU.

The identities pr001 ı pr0000 D pr012 ı pr0000 D pr02 ı pr0000 D pr000 imply that thepull-back of A012 to N0U by pr0000 gives the pairing

.pr0000/�.A012/ W A˝R A! A: (4.2)

The associativity condition implies that the pairing (4.2) endows A with a structure ofa sheaf of associative R-algebras on N0U.


The sheaf Api is endowed with the associative R-algebra structure induced by that

on A. We denote by Apii the A

pi ˝R .Ap

i /op-module A

pi , with the module structure

given by the left and right multiplication.The identities pr201 ı pr1001 D pr000 ı pr10, pr212 ı pr1001 D pr202 ı pr1001 D Id imply

that the pull-back of A012 to N1U by pr1001 gives the pairing

.pr1001/�.A012/ W A1

0 ˝R A01 ! A01: (4.3)

The associativity condition implies that the pairing (4.3) endows A01 with a structureof a A1

0-module. Similarly, the pull-back of A012 to N1U by pr1011 endows A01 witha structure of a .A1

1/op-module. Together, the two module structures define a structure

of a A10 ˝R .A1

1/op-module on A01.

The map (4.1) factors through the map

.pr201/�A01 ˝A2

1.pr212/

�A01 ! .pr202/�A01: (4.4)

Definition 4.6. A unit for a convolution datum A is a morphism of R-modules

1 W R! A

such that the compositions

A011˝Id��! A1

0 ˝R A01

.pr1001

/�.A012/��! A01

and

A01Id˝1��! A01 ˝R A1

1

.pr1011

/�.A012/��! A01

are equal to the respective identity morphisms.

4.2.2 Descent data

Definition 4.7. A descent datum on X is an R-linear convolution datum .U;A/ on Xwith a unit which satisfies the following additional conditions:

1. A01 is locally free of rank one as a A10-module and as a .A1

1/op-module;

2. the map (4.4) is an isomorphism.

4.2.3 1-morphisms. Suppose given convolution data .U;A/ and .U;B/ as in Defi-nition 4.5.

Definition 4.8. A 1-morphism of convolution data

W .U;A/! .U;B/


is a morphism of R-modules 01 W A01 ! B01 such that the diagram

.pr201/�A01 ˝R .pr212/

�A01

A012 ��

.pr201/�.01/˝.pr2

12/�.01/

��

.pr202/�A01

.pr202/�.01/

��

.pr201/�B01 ˝R .pr212/

�B01

B012 �� .pr202/�B01

(4.5)

is commutative.

The 1-morphism induces a morphism ofR-algebras WD .pr000/�.01/ W A! B

onN0U as well as morphisms pi WD .prpi /�./ W Ap

i ! Bpi onNpU. The morphism

01 is compatible with the morphism of algebras 0˝op1 W A1

0˝R .A11/

op ! B10 ˝R

.B11 /

op and the respective module structures.A 1-morphism of descent data is a 1-morphism of the underlying convolution data

which preserves respective units.

4.2.4 2-morphisms. Suppose that we are given descent data .U;A/ and .U;B/ asin 4.2.2 and two 1-morphisms

; W .U;A/! .U;B/

as in 4.2.3.A 2-morphism

b W !

is a section b 2 �.N0UIB/ such that the diagram

A01

01˝b��

b˝01 �� B10 ˝R B01

��

B01 ˝R B11

�� B01

is commutative.

4.2.5 The 2-category of descent data. Fix a cover U of X .Suppose that we are given descent data .U;A/, .U;B/, .U;C/ and 1-morphisms

W .U;A/! .U;B/ and W .U;B/! .U;C/. The map 01 ı 01 W A01 ! C01 isa 1-morphism of descent data ı W .U;A/! .U;C/, the composition of and .

Let .i/ W .U;A/ ! .U;B/, i D 1; 2; 3, be 1-morphisms and let b.j / W .j / !.jC1/, j D 1; 2, be 2-morphisms. The section b.2/ � b.1/ 2 �.N0UIB/ defines a

2-morphism, denoted b.2/b.1/ W .1/ ! .3/, the vertical composition of b.1/ and b.2/.


Suppose that .i/ W .U;A/ ! .U;B/ and .i/ W .U;B/ ! .U;C/, i D 1; 2,

are 1-morphisms and b W .1/ ! .2/ and c W .1/ ! .2/ are 2-morphisms. The

section c � .1/.b/ 2 �.N0UIC/ defines a 2-morphism, denoted c˝ b, the horizontalcomposition of b and c.

We leave it to the reader to check that with the compositions defined above descentdata, 1-morphisms and 2-morphisms form a 2-category, denoted DescR.U/.

4.2.6 Fibered category of descent data. Suppose that � W V ! U is a morphism ofcovers and .U;A/ is a descent datum. Let A

�01 D .N1�/�A01, A

�012 D .N2�/�.A012/.

Then, .V ;A�/ is a descent datum. The assignment .U;A/ 7! .V ;A�/ extends to afunctor, denoted �� W DescR.U/! DescR.V/.

The assignment Cov.X/op 3 U ! DescR.U/, � ! �� is (pseudo-)functor. LetDescR.X/ denote the corresponding 2-category fibered in R-linear 2-categories overCov.X/ with object pairs .U;A/ with U 2 Cov.X/ and .U;A/ 2 DescR.U/; amorphism .U0;A0/ ! .U;A/ in DescR.X/ is a pair .�; /, where � W U0 ! U is amorphism in Cov.X/ and W .U0;A0/! ��.U;A/ D .U0;A�/.

4.3 Trivializations

4.3.1 Definition of a trivialization

Definition 4.9. A trivialization of an algebroid stack C on X is an object in C.X/.

Suppose that C is an algebroid stack on X and L 2 C.X/ is a trivialization. Theobject L determines a morphism EndC .L/

C ! C .

Lemma 4.10. The induced morphism DEndC .L/C ! C is an equivalence.

Remark 4.11. Suppose that C is anR-algebroid stack onX . Then, there exists a coverU of X such that the stack ��0C admits a trivialization.

4.3.2 The 2-category of trivializations. Let TrivR.X/ denote the 2-category with

• objects: the triples .C ;U; L/ where C is an R-algebroid stack on X , U is anopen cover of X such that ��0C.N0U/ is nonempty and L is a trivialization of��0C ,

• 1-morphisms: .C 0;U0; L/! .C ;U; L/ are pairs .F; �/ where � W U0 ! U is amorphism of covers, F W C 0 ! C is a functor such that .N0�/�F.L0/ D L,

• 2-morphisms .F; �/! .G; �/, where .F; �/; .G; �/ W .C 0;U0; L/! .C ;U; L/:the morphisms of functors F ! G.


The assignment .C ;U; L/ 7! C extends in an obvious way to a functor TrivR.X/!AlgStackR.X/.

The assignment .C ;U; L/ 7! U extends to a functor TrivR.X/! Cov.X/makingTrivR.X/ a fibered 2-category over Cov.X/. For U 2 Cov.X/ we denote the fiberover U by TrivR.X/.U /.

4.3.3 Algebroid stacks from descent data. Consider .U;A/ 2 DescR.U/.

The sheaf of algebras A on N0U gives rise to the algebroid stack �AC. The sheafA01 defines an equivalence

01 WD .�/˝A10

A01 W .pr10/� �AC ! .pr11/

� �AC:The convolution map A012 defines an isomorphism of functors

012 W .pr201/�.01/ ı .pr212/

�.01/! .pr202/�.01/:

We leave it to the reader to verify that the triple . �AC; 01; 012/ constitutes a descentdatum for an algebroid stack on X which we denote by St.U;A/.

By construction there is a canonical equivalence �AC ! ��0St.U;A/which endows��0St.U;A/ with a canonical trivialization 1.

The assignment .U;A/ 7! .St.U;A/;U; 1/ extends to a cartesian functor

St W DescR.X/! TrivR.X/:

4.3.4 Descent data from trivializations. Consider .C ;U; L/ 2 TrivR.X/. Since�0 ıpr10 D �0 ıpr11 D �1 we have canonical identifications .pr10/

��0C Š .pr11/��0C Š

��1C . The object L 2 ��0C.N0U/ gives rise to the objects .pr10/�L and .pr11/

�L in��1C.N0U/. Let A01 D Hom��

1C ..pr11/

�L; .pr10/�L/. Thus, A01 is a sheaf of R-

modules on N1U.The object L 2 ��0C.N0U/ gives rise to the objects .pr20/

�L, .pr21/�L and .pr22/

�Lin ��2C.N2U/. There are canonical isomorphisms

.pr2ij /�A01 Š Hom��

2C ..pr2i /

�L; .pr2j /�L/:

The composition of morphisms

Hom��

2C ..pr21/

�L; .pr20/�L/˝R Hom��

2C ..pr22/

�L; .pr21/�L/

! Hom��

2C ..pr22/

�L; .pr20/�L/

gives rise to the map

A012 W .pr201/�A01 ˝R .pr212/

�A01 ! .pr202/�A02:

Since pr1i ıpr000 D Id there is a canonical isomorphism A WD .pr000/�A01 Š End.L/

which supplies A with the unit section 1 W R 1�! End.L/! A.


The pair .U;A/, together with the section 1 is a decent datum which we denotedd.C ;U; L/.

The assignment .U;A/ 7! dd.C ;U; L/ extends to a cartesian functor

dd W TrivR.X/! DescR.X/:

Lemma 4.12. The functors St and dd are mutually quasi-inverse equivalences.

4.4 Base change. For anR-linear category C and homomorphism of algebrasR! S

we denote by C ˝R S the category with the same objects as C and morphisms definedby HomC˝RS .A;B/ D HomC .A;B/˝R S .

For an R-algebra A the categories .A ˝R S/C and AC ˝R S are canonicallyisomorphic.

For a prestack C inR-linear categories we denote by C˝RS the prestack associatedto the fibered category U 7! C.U /˝R S .

ForU � X ,A;B 2 C.U /, there is an isomorphism of sheaves HomC˝RS .A;B/ DHomC .A;B/˝R S .

Lemma 4.13. Suppose that A is a sheaf ofR-algebras and C is anR-algebroid stack.

1. . �AC ˝R S/z is an algebroid stack equivalent to E.A˝R S/C .

2. BC ˝R S is an algebroid stack.

Proof. Suppose that A is a sheaf of R-algebras. There is a canonical isomorphismof prestacks .A ˝R S/C Š AC ˝R S which induces the canonical equivalenceE.A˝R S/C ŠDAC ˝R S .

The canonical functor AC ! �AC induces the functor AC ˝R S ! �AC ˝R S ,

hence the functor DAC ˝R S ! . �AC ˝R S/z.The map A ! A ˝R S induces the functor �AC ! E.A˝R S/C which factors

through the functor �AC ˝R S ! E.A˝R S/C. From this we obtain the functor

. �AC ˝R S/z !E.A˝R S/C.We leave it to the reader to check that the two constructions are mutually in-

verse equivalences. It follows that . �AC ˝R S/z is an algebroid stack equivalent toE.A˝R S/C.

Suppose that C is an R-algebroid stack. Let U be a cover such that ��0C.N0U/ isnonempty. Let L be an object in ��0C.N0U/; put A WD End��

0C .L/. The equivalence

�AC ! ��0C induces the equivalence . �AC˝R S/z ! .��0C ˝R S/z. Since the formeris an algebroid stack so is the latter. There is a canonical equivalence .��0C ˝R S/z Š��0 . BC ˝R S/; since the former is an algebroid stack so is the latter. Since the property

of being an algebroid stack is local, the stack BC ˝R S is an algebroid stack.


4.5 Twisted forms. Suppose that A is a sheaf of R-algebras on X . We will call an

R-algebroid stack locally equivalent to AAopC a twisted form of A.Suppose that S is twisted form of OX . Then, the substack iS is an O�X -gerbe.

The assignment S 7! iS extends to an equivalence between the 2-groupoid of twistedforms of OX (a subcategory of AlgStackC.X/) and the 2-groupoid of O�X -gerbes.

Let S be a twisted form of OX . Then for any U � X , A 2 S.U / the canonicalmap OU ! EndS .A/ is an isomorphism. Consequently, if U is a cover of X and Lis a trivialization of ��0S , then there is a canonical isomorphism of sheaves of algebrasON0U ! End��

0S .L/.

Conversely, suppose that .U;A/ is a C-descent datum. If the sheaf of algebrasA is isomorphic to ON0U then such an isomorphism is unique since the latter has nonon-trivial automorphisms. Thus, we may and will identify A with ON0U. Hence, A01

is a line bundle on N1U and the convolution map A012 is a morphism of line bundles.The stack which corresponds to .U;A/ (as in 4.3.3) is a twisted form of OX .

Isomorphism classes of twisted forms of OX are classified by H 2.X IO�X /. Werecall the construction presently. Suppose that the twisted form S of OX is representedby the descent datum .U;A/. Assume in addition that the line bundle A01 on N1U istrivialized. Then we can consider A012 as an element in�.N2UIO�/. The associativitycondition implies that A012 is a cocycle in LC 2.UIO�/. The class of this cocycle inLH 2.UIO�/ does not depend on the choice of trivializations of the line bundle A01 and

yields a class in H 2.X IO�X /.We can write this class using the de Rham complex for jets. We refer to Section 7.1

for the notations and a brief review.

The composition O� ! O�=C�log�! O=C

j1

��! DR.J=O/ induces the mapH 2.X IO�/! H 2.X IDR.J=O// Š H 2.�.X I��

X˝JX=OX /;rcan/. Here the latterisomorphism follows from the fact that the sheaf ��

X ˝ JX=OX is soft. We denote byŒS � 2 H 2.�.X I��

X ˝ JX=OX /;rcan/ the image of the class of S in H 2.X IO�/. InLemma 7.13 (see also Lemma 7.15) we will construct an explicit representative for ŒS �.

5 DGLA of local cochains on matrix algebras

In this section we define matrix algebras from a descent datum and use them to con-struct a cosimplicial DGLA of local cochains. We also establish the acyclicity of thiscosimplicial DGLA.

5.1 Definition of matrix algebras

5.1.1 Matrix entries. Suppose that .U;A/ is an R-descent datum. Let A10 WD��A01, where � D pr110 W N1U! N1U is the transposition of the factors. The pairings.pr1100/

�.A012/ W A10 ˝R A10 ! A10 and .pr1110/

�.A012/ W A11 ˝R A10 ! A10 of

sheaves on N1U endow A10 with a structure of a A11 ˝ .A1

0/op-module.


The identities pr201 ı pr1010 D Id, pr212 ı pr1010 D � and pr202 ı pr1010 D pr000 ı pr10imply that the pull-back of A012 by pr1010 gives the pairing

.pr1010/�.A012/ W A01 ˝R A10 ! A1

00: (5.1)

which is a morphism of A10 ˝K .A1

0/op-modules. Similarly, we have the pairing

.pr1101/�.A012/ W A10 ˝R A01 ! A1

11: (5.2)

The pairings (4.2), (4.3), (5.1) and (5.2), Aij ˝R Ajk ! Aik are morphisms ofA1i ˝R .A1

k/op-modules which, as a consequence of associativity, factor through maps

Aij ˝A1j

Ajk ! Aik (5.3)

induced by Aijk D .pr1ijk/�.A012/; here i; j; k D 0; 1. Define now for every p � 0

the sheaves Apij , 0 � i; j � p, on NpU by A

pij D .prpij /

�A01. Define also Ap

ijkD

.prpijk/�.A012/. We immediately obtain for every p the morphisms

Ap

ijkW Ap

ij ˝Ap

jAp

jk! A

p

ik: (5.4)

5.1.2 Matrix algebras. Let Mat.A/0 D A; thus, Mat.A/0 is a sheaf of algebras onN0U. For p D 1; 2; : : : let Mat.A/p denote the sheaf on NpU defined by

Mat.A/p DpM

i;jD0Apij :

The maps (5.4) define the pairing

Mat.A/p ˝Mat.A/p ! Mat.A/p

which endows the sheaf Mat.A/p with a structure of an associative algebra by virtueof the associativity condition. The unit section 1 is given by 1 DPp

iD0 1i i , where 1i iis the image of the unit section of A

pii .

5.1.3 Combinatorial restriction. The algebras Mat.A/p , p D 0; 1; : : :, do not forma cosimplicial sheaf of algebras on NU in the usual sense. They are, however, relatedby combinatorial restriction which we describe presently.

For a morphism f W Œp�! Œq� in � define a sheaf on NqU by

f ]Mat.A/q DpM

i;jD0Aq

f .i/f .j /:

Note that f ]Mat.A/q inherits a structure of an algebra.Recall from Section 2.3.1 that the morphism f induces the map f � W NqU! NpU

and that f� denotes the pull-back along f �. We will also use f� to denote the canonicalisomorphism of algebras

f� W f�Mat.A/p ! f ]Mat.A/q (5.5)

induced by the isomorphisms f�Apij Š A

q

f .i/f .j /.


5.1.4 Refinement. Suppose that � W V ! U is a morphism of covers. For p D0; 1; : : : there is a natural isomorphism

.Np�/�Mat.A/p Š Mat.A�/p

of sheaves of algebras on NpV . The above isomorphisms are obviously compatiblewith combinatorial restriction.

5.2 Local cochains on matrix algebras

5.2.1 Local cochains. A sheaf of matrix algebras is a sheaf of algebras B togetherwith a decomposition

B DpM

i;jD0Bij

as a sheaf of vector spaces which satisfies Bij �Bjk D Bik .To a matrix algebra B one can associate to the DGLA of local cochains defined as

follows. For n D 0 let C 0.B/loc DLBi i B D C 0.B/. For n > 0 let C n.B/loc

denote the subsheaf of C n.B/ whose stalks consist of multilinear maps D such thatfor any collection of sikjk 2 Bikjk

1. D.si1j1 ˝ � � � ˝ sinjn/ D 0 unless jk D ikC1 for all k D 1; : : : ; n � 1,

2. D.si0i1 ˝ si1i2 ˝ � � � ˝ sin�1in/ 2 Bi0in .

For I D .i0; : : : ; in/ 2 Œp��nC1 let

C I .B/loc WD Homk.˝n�1jD0Bij ijC1;Bi0in/:

The restriction maps along the embeddings ˝n�1jD0Bij ijC1,! B˝n induce an isomor-

phism C n.B/loc !˚I2Œp��nC1C I .B/loc.The sheafC �.B/locŒ1� is a subDGLA ofC �.B/Œ1� and the inclusion map is a quasi-

isomorphism.For a matrix algebra B on X we denote by Def.B/loc.R/ the subgroupoid of

Def.B/.R/ with objects R-star products which respect the decomposition given by.B˝kR/ij D Bij˝kR and 1- and 2-morphisms defined accordingly. The composition

Def.B/loc.R/! Def.B/.R/! MC2.�.X IC �

.B/Œ1�/˝k mR/

takes values in MC2.�.X IC �.B/locŒ1�/ ˝k mR/ and establishes an isomorphism of2-groupoids Def.B/loc.R/ Š MC2.�.X IC �.B/locŒ1�/˝k mR/.

5.2.2 Combinatorial restriction of local cochains. Suppose given a matrix algebraB DLq

i;jD0 Bij is a sheaf of matrix k-algebras.


The DGLAC �.B/locŒ1�has additional variance not exhibited byC �.B/Œ1�. Namely,for f W Œp�! Œq� – a morphism in � – there is a natural map of DGLA

f ] W C �

.B/locŒ1�! C�

.f ]B/locŒ1� (5.6)

defined as follows. Let f ij]W .f ]B/ij ! Bf .i/f .j / denote the tautological isomor-

phism. For each multi-index I D .i0; : : : ; in/ 2 Œp��nC1 let

f I] WD ˝n�1jD0fij ijC1

]W ˝n�1jD0 .f ]B/ij ijC1

!˝n�1iD0Bf .ij /f .ijC1/:

Let f n]WD ˚

I2†�nC1p

f I]

. The map (5.6) is defined as restriction along f n]

.

Lemma 5.1. The map (5.6) is a morphism of DGLA

f ] W C �

.B/locŒ1�! C�

.f ]B/locŒ1�:

If follows that combinatorial restriction of local cochains induces the functor

MC2.f ]/ W MC2.�.X IC �

.B/locŒ1�/˝kmR/! MC2.�.X IC �

.f ]B/locŒ1�/˝kmR/:

Combinatorial restriction with respect to f induces the functor

f ] W Def.B/loc.R/! Def.f ]B/loc.R/:

It is clear that the following diagram commutes:

Def.B/loc.R/f ]

��

��

Def.f ]B/loc.R/

��

MC2.�.X IC �.B/locŒ1�/˝k mR/MC2.f ]/

�� MC2.�.X IC �.f ]B/locŒ1�/˝k mR/.

5.2.3 Cosimplicial DGLA from descent datum. Suppose that .U;A/ is a descentdatum for a twisted sheaf of algebras as in 4.2.2. Then, for each p D 0; 1; : : : wehave the matrix algebra Mat.A/p as defined in 5.1, and therefore the DGLA of localcochains C �.Mat.A/p/locŒ1� defined in 5.2.1. For each morphism f W Œp�! Œq� thereis a morphism of DGLA

f ] W C �

.Mat.A/q/locŒ1�! C�

.f ]Mat.A/q/locŒ1�

and an isomorphism of DGLA

C�

.f ]Mat.A/q/locŒ1� Š f�C �

.Mat.A/p/locŒ1�

induced by the isomorphism f� W f�Mat.A/p ! f ]Mat.A/q from the equation (5.5).These induce the morphisms of the DGLA of global sections

�.NqUIC �.Mat.A/q/locŒ1�/

f ]

��.NpUIC �.Mat.A/p/locŒ1�/

f�

��

�.NqUIf�C �.Mat.A/p/locŒ1�/.


For � W Œn�! � let

G.A/� D �.N�.n/UI�.0n/�C �

.Mat.A/�.0//locŒ1�/:

Suppose given another simplex � W Œm� ! � and morphism W Œm� ! Œn� suchthat � D � ı (i.e. is a morphism of simplices �! �). The morphism .0n/ factorsuniquely into 0 ! .0/ ! .m/ ! n, which, under �, gives the factorization of�.0n/ W �.0/! �.n/ (in �) into

�.0/f��! �.0/

g��! �.m/h�! �.n/; (5.7)

where g D �.0m/. The map

� W G.A/� ! G.A/�

is the composition

�.N�.m/UIg�C �

.Mat.A/�.0//locŒ1�/

h��! �.N�.n/UI h�g�C �

.Mat.A/�.0//locŒ1�/

f ]��! �.N�.n/UI h�g�f�C �

.Mat.A/�.0//locŒ1�/:

Suppose given yet another simplex, � W Œl � ! �, and a morphism of simplices W � ! �, i.e. a morphism W Œl �! Œm� such that � D � ı . Then, the composition� ı � W G.A/� ! G.A/� coincides with the map . ı /�.

For n D 0; 1; 2; : : : let

G.A/n DY

Œn��!�

G.A/�: (5.8)

A morphism W Œm� ! Œn� in � induces the map of DGLA � W G.A/m ! G.A/n.The assignment � 3 Œn� 7! G.A/n, 7! � defines the cosimplicial DGLA denotedby G.A/.

5.3 Acyclicity

Theorem 5.2. The cosimplicial DGLA G.A/ is acyclic, i.e. it satisfies the condition(3.8).

The rest of the section is devoted to the proof of Theorem 5.2. We fix a degree ofHochschild cochains k.

For � W Œn�! � let c� D �.N�.n/UI�.0n/�C k.Mat.A/�.0//loc/. For a morphism W �! � we have the map � W c� ! c� defined as in 5.2.3.

Let .C�; @/ denote the corresponding cochain complex whose definition we recallbelow. For n D 0; 1; : : : let Cn D Q

Œn��!�

c�. The differential @n W Cn ! CnC1 is

defined by the formula @n DPnC1iD0 .�1/i .@ni /�.


5.3.1 Decomposition of local cochains. As was noted in 5.2.1, for n; q D 0; 1; : : :

there is a direct sum decomposition

C k.Mat.A/q/loc DM

I2Œq�kC1

C I .Mat.A/q/loc: (5.9)

In what follows we will interpret a multi-index I D .i0; : : : ; in/ 2 Œq�kC1 as a mapI W f0; : : : ; kg ! Œq�. For I as above let s.I / D j Im.I /j � 1. The map I factorsuniquely into the composition

f0; : : : ; kg I0

�! Œs.I /�m.I/��! Œq�

where the second map is a morphism in� (i.e. is order preserving). Then, the isomor-phisms m.I/�AI 0.i/I 0.j / Š AI.i/I.j / induce the isomorphism

m.I/�C I0

.Mat.A/s.I //loc ! C I .Mat.A/q/loc:

Therefore, the decomposition (5.9) may be rewritten as follows:

C k.Mat.A/q/loc DMe

MI

e�C I .Mat.A/p/loc (5.10)

where the summation is over injective (monotone) maps e W Œs.e/�! Œq� and surjectivemaps I W f0; : : : ; kg ! Œs.e/�. Note that, for e, I as above, there is an isomorphism

e�C I .Mat.A/p/loc Š C eBI .e]Mat.A/s.e//loc:

5.3.2 Filtrations. Let F �C k.Mat.A/q/loc denote the decreasing filtration defined by

F sC k.Mat.A/q/loc DM

eWs.e/�s

MI

e�C I .Mat.A/p/loc

Note that F sC k.Mat.A/q/loc D 0 for s > k and GrsC k.Mat.A/q/loc D 0 for s < 0.The filtration F �C k.Mat.A/�.0//loc induces the filtration F �c�, hence the filtration

F �C� withF sc�D�.N�.n/UI�.0n/�F sC k.Mat.A/�.0//loc/,F sCnDQŒn�

��!�F sc�.

The following result then is an easy consequence of the definitions:

Lemma 5.3. For W �! � the induced map � W c� ! c� preserves filtration.

Corollary 5.4. The differential @n preserves the filtration.

Proposition 5.5. For any s the complex Grs.C�; @/ is acyclic in non-zero degrees.

Proof. Use the following notation: for a simplex � W Œm�! � and an arrow e W Œs�!Œ�.0/� the simplex �e W Œm C 1� ! � is defined by �e.0/ D Œs�, �e.01/ D e, and�e.i/ D �.i � 1/, �e.i; j / D �.i � 1; j � 1/ for i > 0.


For D D .D�/ 2 F sCn, D� 2 c� and � W Œn � 1�! � let

hn.D/� DXI;e

e�DI�e

where e W Œs�! �.0/ is an injective map, I W f0; : : : ; kg ! Œs� is a surjection, andDI�e

is the I -component of D�e in the sense of the decomposition (5.9). Put hn.D/ D.hn.D/�/ 2 Cn�1. It is clear that hn.D/ 2 F sCn�1. Thus the assignment D 7!hn.D/ defines a filtered map hn W Cn ! Cn�1 for all n � 1. Hence, for any s we havethe map Grshn W GrsCn ! GrsCn�1.

Next, we calculate the effect of the face maps @n�1i W Œn � 1� ! Œn�. First of all,note that, for 1 � i � n� 1, .@n�1i /� D Id. The effect of .@n�10 /� is the map on globalsections induced by

�.01/] W �.1n/�C k.Mat.A/�.1//loc ! �.0n/�C k.Mat.A/�.0//loc

and .@n�1n /� D �.n � 1; n/�. Thus, for D D .D�/ 2 F sCn, we have:

..@n�1i /�hn.D//� D .@n�1i /�XI;e

e�DI

.�ı@n�1i

/e

D

8<ˆ:

�.01/]PI;e e�DI

.�ı@n�10

/eif i D 0,P

I;e e�DI

.�ı@n�1i

/eif 1 � i � n � 1,

�.n � 1; n/�PI;e e�DI

.�ı@n�1n /e

if i D n.

On the other hand,

hnC1..@nj /�D/� DXI;e

e�..@nj /�D/I�e

D

8ˆ<ˆ:

PI;e e�e]DI

�if j D 0,P

I;e e�DI

.�ı@n�10

/�.01/ıeif j D 1,P

I;e e�DI

.�ı@n�1j�1

/eif 2 � j � n,P

I;e �.n � 1; n/�e�DI

.�ı@n�1n /e

if j D nC 1.

Note thatPI;e e�e]DI

�D D� mod F sC1c�, i.e. the map hnC1 ı @n0 induces the

identity map on Grsc� for all �, hence the identity map on GrsCn.The identities .@n�1i hn.D//� D hnC1..@niC1/�D/� mod F sc� hold for0 � i � n.For n D 0; 1; : : : let @n D P

i .�1/@ni . The above identities show that, for n > 0,hnC1 ı @n C @n�1 ı hn D Id.

Proof of Theorem 5.2. Consider for a fixed k the complex of Hochschild degree kcochains C�. It is shown in Proposition 5.5 that this complex admits a finite filtrationF �C� such that GrC� is acyclic in positive degrees. Therefore C�is also acyclic inpositive degree. As this holds for every k, the condition (3.8) is satisfied.


6 Deformations of algebroid stacks

In this section we define a 2-groupoid of deformations of an algebroid stack. We alsodefine 2-groupoids of deformations and star products of a descent datum and relate itwith the 2-groupoid G-stacks of an appropriate cosimplicial DGLA.

6.1 Deformations of linear stacks

Definition 6.1. Let B be a prestack onX inR-linear categories. We say that B is flat iffor anyU � X ,A;B 2 B.U / the sheaf HomB.A;B/ is flat (as a sheaf ofR-modules).

Suppose now that C is a stack in k-linear categories on X and R is a commutativeArtin k-algebra. We denote by Def.C/.R/ the 2-category with

• objects: pairs .B;$/, where B is a stack in R-linear categories flat over R and$ W BB ˝R k ! C is an equivalence of stacks in k-linear categories;

• 1-morphisms: a 1-morphism .B.1/;$ .1//! .B.2/;$ .2// is a pair .F; �/whereF W B.1/ ! B.2/ is a R-linear functor and � W $ .2/ ı .F ˝R k/ ! $ .1/ is anisomorphism of functors;

• 2-morphisms: a 2-morphism .F 0; � 0/ ! .F 00; � 00/ is a morphism of R-linearfunctors � W F 0 ! F 00 such that � 00 ı .Id$.2/ ˝ .� ˝R k// D � 0.

The 2-category Def.C/.R/ is a 2-groupoid.

Lemma 6.2. Suppose that B is a flat R-linear stack on X such that BB ˝R k is analgebroid stack. Then, B is an algebroid stack.

Proof. Let x 2 X . Suppose that for any neighborhood U of x the category B.U / isempty. Then, the same is true about BB ˝R k.U /which contradicts the assumption thatBB ˝R k is an algebroid stack. Therefore, B is locally nonempty.

Suppose that U is an open subset and A;B are two objects in B.U /. Let xAand xB be their respective images in BB ˝R k.U /. We have: Hom

AB˝Rk.U /.xA; xB/ D

�.U IHomB.A;B/˝R k/. Replacing U by a subset we may assume that there is anisomorphism x W xA! xB .

The short exact sequence

0! mR ! R! k ! 0

gives rise to the sequence

0! HomB.A;B/˝R mR ! HomB.A;B/! HomB.A;B/˝R k ! 0

of sheaves on U which is exact due to flatness of HomB.A;B/. The surjectivity of themap HomB.A;B/ ! HomB.A;B/ ˝R k implies that for any x 2 U there exists aneighborhood x 2 V � U and 2 �.V IHomB.AjV ; BjV // D HomB.V /.AjV ; BjV /such that xjV is the image of . Since x is an isomorphism and mR is nilpotent itfollows that is an isomorphism.


6.2 Deformations of descent data. Suppose that .U;A/ is an k-descent datum andR is a commutative Artin k-algebra.

We denote by Def 0.U;A/.R/ the 2-category with

• objects: R-deformations of .U;A/; such a gadget is a flat R-descent datum.U;B/ together with an isomorphism of k-descent data W .U;B ˝R k// !.U;A/;

• 1-morphisms: a 1-morphism of deformations .U;B.1/; .1//! .U;B.2/; .2//

is a pair .; a/, where W .U;B.1//! .U;B.2// is a 1-morphism of R-descent

data and a W .2/ ı . ˝R k/! .1/ is 2-isomorphism;

• 2-morphisms: a 2-morphism .0; a0/ ! .00; a00/ is a 2-morphism b W 0 ! 00such that a00 ı .Id .2/ ˝ .b ˝R k// D a0.

Suppose that .; a/ is a 1-morphism. It is immediate from the definition above thatthe morphism of k-descent data ˝R k is an isomorphism. SinceR is an extension ofk by a nilpotent ideal the morphism is an isomorphism. Similarly, any 2-morphismis an isomorphism, i.e. Def 0.U;A/.R/ is a 2-groupoid.

The assignmentR 7! Def 0.U;A/.R/ is fibered 2-category in 2-groupoids over thecategory of commutative Artin k-algebras ([3]).

6.2.1 Star products. An (R-)star product on .U;A/ is a deformation .U;B; / of.U;A/ such that B01 D A01 ˝k R and W B01 ˝R k ! A01 is the canonicalisomorphism. In other words, a star product is a structure of an R-descent datum on.U;A˝k R/ such that the natural map

.U;A˝k R/! .U;A/

is a morphism of such.We denote by Def.U;A/.R/ the full 2-subcategory of Def 0.U;A/.R/ whose ob-

jects are star products.The assignment R 7! Def.U;A/.R/ is fibered 2-category in 2-groupoids over the

category of commutative Artin k-algebras ([3]) and the inclusions Def.U;A/.R/ !Def 0.U;A/.R/ extend to a morphism of fibered 2-categories.

Proposition 6.3. Suppose that .U;A/ is a C-linear descent datum with A D ON0U.Then, the embedding Def.U;A/.R/! Def 0.U;A/.R/ is an equivalence.

6.2.2 Deformations and G -stacks. Suppose that .U;A/ is a k-descent datum and.U;B/ is anR-star product on .U;A/. Then, for everyp D 0; 1; : : : the matrix algebraMat.B/p is a flat R-deformation of the matrix algebra Mat.A/p . The identificationB01 D A01 ˝k R gives rise to the identification Mat.B/p D Mat.A/p ˝k R ofthe underlying sheaves of R-modules. Using this identification we obtain the Maurer–Cartan element�p 2 �.NpUIC 2.Mat.A/p/loc˝kmR/. Moreover, the equation (5.5)


implies that for a morphism f W Œp�! Œq� in�we have f��p D f ]�q . Therefore thecollection �p defines an element in Stackstr.G.A/ ˝k mR/0. The considerations in5.2.1 and 5.2.2 imply that this construction extends to an isomorphism of 2-groupoids

Def.U;A/.R/! Stackstr.G.A/˝k mR/: (6.1)

Combining (6.1) with the embedding

Stackstr.G.A/˝k mR/! Stack.G.A/˝k mR/ (6.2)

we obtain the functor

Def.U;A/.R/! Stack.G.A/˝k mR/: (6.3)

The naturality properties of (6.3) with respect to base change imply that (6.3) extendsto morphism of functors on the category of commutative Artin algebras.

Combining this with the results of Theorems 5.2 and 3.12 implies the following:

Proposition 6.4. The functor (6.3) is an equivalence.

Proof. By Theorem 5.2 the DGLA G.A/ ˝k mR satisfies the assumptions of Theo-rem 3.12. The latter says that the inclusion (6.2) is an equivalence. Since (6.1) is anisomorphism, the composition (6.3) is an equivalence as claimed.

7 Jets

In this section we use constructions involving the infinite jets to simplify the cosimplicialDGLA governing the deformations of a descent datum.

7.1 Infinite jets of a vector bundle. LetM be a smooth manifold, and E a locally-freeOM -module of finite rank.

Let i W M �M ! M , i D 1; 2, denote the projection on the i th factor. Denoteby �M W M ! M �M the diagonal embedding and let ��M W OM�M ! OM be theinduced map. Let IM WD ker.��M /.

LetJk.E/ WD .1/�

�OM�M=IkC1M ˝ �1

2OM

�12 E�;

JkM WD Jk.OM /. It is clear from the above definition that JkM is, in a natural way, asheaf of commutative algebras and Jk.E/ is a sheaf of JkM -modules. If moreover E isa sheaf of algebras, Jk.E/ will canonically be a sheaf of algebras as well. We regardJk.E/ as OM -modules via the pull-back map �1 W OM ! .1/�OM�M

For 0 � k � l the inclusion IlC1M ! IkC1M induces the surjective map Jl.E/ !Jk.E/. The sheaves Jk.E/, k D 0; 1; : : : together with the maps just defined form aninverse system. Define J.E/ WD lim �Jk.E/. Thus, J.E/ carries a natural topology.

We denote by pE W J.E/! E the canonical projection. In the case when E D OMwe denote by p the corresponding projection p W JM ! OM . By j k W E ! Jk.E/ we


denote the map e 7! 1˝ e, and j1 WD lim �jk . In the case E D OM we also have the

canonical embedding OM ! JM given by f 7! f � j1.1/.Let

d1 W OM�M ˝ �

2OM

�12 E ! �11 �1M ˝ �11

OMOM�M ˝ �1

2OM

�12 E

denote the exterior derivative along the first factor. It satisfies

d1.IkC1M ˝ �1

2OM

�12 E/ �11 �1M ˝ �11

OMIkM ˝ �1

2OM

�12 E

for each k and, therefore, induces the map

d.k/1 W Jk.E/! �1M=P ˝OM Jk�1.E/:

The maps d .k/1 for different values of k are compatible with the maps Jl.E/! Jk.E/

giving rise to the canonical flat connection

rcan W J.E/! �1M ˝ J.E/:

Here and below we use notation .�/˝ J.E/ for lim �.�/˝OM Jk.E/.

Sincercan is flat we obtain the complex of sheaves DR.J.E//D .��

M˝J.E/;rcan/.When E D OM embedding OM ! JM induces embedding of de Rham complexDR.O/ D .��

M ; d / into DR.J/. We denote the quotient by DR.J=O/. All the com-plexes above are complexes of soft sheaves. We have the following:

Proposition 7.1. The (hyper)cohomology H i .M;DR.J.E/// Š H i .�.M I�M ˝J.E//;rcan/ is 0 if i > 0. The map j1 W E ! J.E/ induces the isomorphism between�.E/ and H 0.M;DR.J.E/// Š H 0.�.M I�M ˝ J.E//;rcan/

7.2 Jets of line bundles. Let, as before,M be a smooth manifold, JM be the sheaf ofinfinite jets of smooth functions onM and p W JM ! OM be the canonical projection.Set JM;0 D ker p. Note that JM;0 is a sheaf of OM modules and therefore is soft.

Suppose now that L is a line bundle onM . Let Isom0.L˝JM ;J.L// denote thesheaf of local JM -module isomorphisms L ˝ JM ! J.L/ such that the followingdiagram is commutative:

L˝ JM ��

Id˝p��

�� J.L/

pL��

��

��

L.

It is easy to see that the canonical map JM ! EndJM.L˝JM / is an isomorphism.

For 2 JM;0 the exponential series exp./ converges. The composition

JM;0exp��! JM ! EndJM

.L˝ JM /


defines an isomorphism of sheaves of groups

exp W JM;0 ! Aut0.L˝ JM /;

where Aut0.L˝ JM / is the sheaf of groups of (locally defined) JM -linear automor-phisms of L˝ JM making the diagram

L˝ JM ��

Id˝p��

��L˝ JM

Id˝p��

L

commutative.

Lemma 7.2. The sheaf Isom0.L˝ JM ;J.L// is a torsor under the sheaf of groupsexp JM;0.

Proof. Since L is locally trivial, both J.L/ and L˝JM are locally isomorphic to JM .Therefore the sheaf Isom0.L˝ JM ;J.L// is locally non-empty, hence a torsor.

Corollary 7.3. The torsor Isom0.L˝ JM ;J.L// is trivial, that is, Isom0.L˝ JM ;

J.L// WD �.M I Isom0.L˝ JM ;J.L/// ¤ ¿.

Proof. Since the sheaf of groups JM;0 is soft we have H 1.M;JM;0/ D 0 (see [8],Lemme 22, cf. also [6], Proposition 4.1.7). Therefore every JM;0-torsor is trivial.

Corollary 7.4. The set Isom0.L˝ JM ;J.L// is an affine space with the underlyingvector space �.M IJM;0/.

Let L1 and L2 be two line bundles, and f W L1 ! L2 an isomorphism. Thenf induces a map Isom0.L2 ˝ JM ;J.L2// ! Isom0.L1 ˝ JM ;J.L1// which wedenote by Ad f :

Ad f .�/ D .j1.f //�1 ı � ı .f ˝ Id/:

Let L be a line bundle on M and f W N ! M is a smooth map. Then there is apull-back map f � W Isom0.L˝ JM ;J.L//! Isom0.f

�L˝ JN ;J.f�L//.

If L1, L2 are two line bundles, and �i 2 Isom0.Li ˝JM ;J.Li //, i D 1; 2. Thenwe denote by �1˝ �2 the induced element of Isom0..L1˝L2/˝JM ;J.L1˝L2//.

For a line bundle L let L� be its dual. Then given � 2 Isom0.L ˝ JM ;J.L//

there exists a unique a unique �� 2 Isom0.L�˝ JM ;J.L

�// such that ��˝ � D Id.For any bundle E J.E/ has a canonical flat connection which we denote by rcan .

A choice of � 2 Isom0.L˝ JM ;J.L// induces the flat connection ��1 ı rcanLı � on

L˝ JM .Let r be a connection on L with the curvature � . It gives rise to the connection

r ˝ IdC Id˝rcan on L˝ JM .


Lemma 7.5. 1. Choose � , r as above. Then the difference

F.�;r/ D ��1 ı rcanL ı � � .r ˝ IdC Id˝rcan/ (7.1)

is an element of 2 �1 ˝ EndJM.L˝ JM / Š �1 ˝ JM .

2. Moreover, F satisfies

rcanF.�;r/C � D 0: (7.2)

Proof. We leave the verification of the first claim to the reader. The flatness of ��1 ırcan

Lı � implies the second claim.

The following properties of our construction are immediate

Lemma 7.6. 1. Let L1 and L2 be two line bundles, andf W L1 ! L2 an isomorphism.Let r be a connection on L2 and � 2 Isom0.L2 ˝ JM ;J.L2//. Then

F.�;r/ D F.Ad f .�/;Ad f .r//:2. Let L be a line bundle on M , r a connection on L and � 2 Isom0.L˝ JM ;

J.L//. Let f W N !M be a smooth map. Then

f �F.�;r/ D F.f ��; f �r/:3. Let L be a line bundle on M , r a connection on L and � 2 Isom0.L˝ JM ;

J.L//. Let 2 �.M IJM;0/. Then

F. � �;r/ D F.�;r/Crcan:

4. Let L1, L2 be two line bundles with connections r1 and r2 respectively, andlet �i 2 Isom0.Li ˝ JM ;J.Li //, i D 1; 2. Then

F.�1 ˝ �2;r1 ˝ IdC Id˝r2/ D F.�1;r1/C F.�2;r2/:

7.3 DGLAs of infinite jets. Suppose that .U;A/ is a descent datum representing atwisted form of OX . Thus, we have the matrix algebra Mat.A/ and the cosimplicialDGLA G.A/ of local C-linear Hochschild cochains.

The descent datum .U;A/ gives rise to the descent datum .U;J.A//, J.A/ D.J.A/;J.A01/; j

1.A012//, representing a twisted form of JX , hence to the matrixalgebra Mat.J.A// and the corresponding cosimplicial DGLA G.J.A// of local O-linear continuous Hochschild cochains.

The canonical flat connection rcan on J.A/ induces the flat connection, still de-noted rcan on Mat.J.A//p for each p; the product on Mat.J.A//p is horizontal withrespect to rcan. The flat connection rcan induces the flat connection, still denotedrcan on C �.Mat.J.A//p/locŒ1� which acts by derivations of the Gerstenhaber bracketand commutes with the Hochschild differential ı. Therefore we have the sheaf of


DGLA DR.C �.Mat.J.A//p/locŒ1�/ with the underlying sheaf of graded Lie algebras��

NpU˝ C �.Mat.J.A//p/locŒ1� and the differential rcan C ı.


GDR.J.A//� D �.N�.n/UI�.0n/�DR.C �

.Mat.J.A//�.0//locŒ1�//

be the DGLA of global sections. The “inclusion of horizontal sections” map inducesthe morphism of DGLA

j1 W G.A/� ! GDR.J.A//�:

For W Œm�! Œn� in �, � D � ı there is a morphism of DGLA

� W GDR.J.A//� ! GDR.J.A//

�

making the diagram

G.A/�j1

��

�

��

GDR.J.A//�

�

��

G.A/�j1

�� GDR.J.A//�

commutative.Let GDR.J.A//

n D QŒn�

��!�GDR.J.A//

�. The cosimplicial DGLA GDR.J.A//

is defined by the assignment � 3 Œn� 7! GDR.J.A//n, 7! �.

Proposition 7.7. The map j1 W G.A/ ! G.J.A// extends to a quasiisomorphismof cosimplicial DGLA.

j1 W G.A/! GDR.J.A//:

The goal of this section is to construct a quasiisomorphism of the latter DGLA withthe simpler DGLA.

The canonical flat connection rcan on JX induces a flat connection on xC �.JX /Œ1�,the complex of O-linear continuous normalized Hochschild cochains, still denotedrcan which acts by derivations of the Gerstenhaber bracket and commutes with theHochschild differential ı. Therefore we have the sheaf of DGLA DR. xC �.JX /Œ1�/ withthe underlying graded Lie algebra ��

X ˝ xC �.JX /Œ1� and the differential ı Crcan.Recall that the Hochschild differential ı is zero on C 0.JX / due to commutativity

of JX . It follows that the action of the sheaf of abelian Lie algebras JX D C 0.JX /

on xC �.JX /Œ1� via the restriction of the adjoint action (by derivations of degree �1)commutes with the Hochschild differential ı. Since the cochains we consider areOX -linear, the subsheaf OX D OX � j1.1/ JX acts trivially. Hence the action ofJX descends to an action of the quotient JX=OX . This action induces the action ofthe abelian graded Lie algebra��

X ˝JX=OX by derivations on the graded Lie algebra��

X˝ xC �.JX /Œ1�. Further, the subsheaf .��

X˝JX=OX /cl WD ker.rcan/ acts by deriva-

tion which commute with the differential ı Crcan. For ! 2 �.X I .�2X ˝ JX=OX /cl/


we denote by DR. xC �.JX /Œ1�/! the sheaf of DGLA with the underlying graded Liealgebra ��

X ˝ xC �.JX /Œ1� and the differential ı Crcan C �! . Let

gDR.JX /! D �.X IDR. xC �

.JX /Œ1�/!/;

be the corresponding DGLA of global sections.Suppose now that U is a cover of X ; let � W NU ! X denote the canonical map.


GDR.J/�! D �.N�.n/UI ��DR. xC �

.JX /Œ1�/!/:

For � W Œm� ! � and a morphism W Œm� ! Œn� in � such that � D � ı the map�.m/! �.n/ induces the map

� W GDR.J/�! ! GDR.J/

�! :

For n D 0; 1; : : : let GDR.J/n! D

QŒn�

��!�GDR.J/

�! . The assignment Œn� 7! GDR.J/

n!

extends to a cosimplicial DGLA GDR.J/! . If we need to explicitly indicate the coverwe will also denote this DGLA by GDR.J/!.U/.

Lemma 7.8. The cosimplicial DGLA GDR.J/! is acyclic, i.e. satisfies the condi-tion (3.8).

Proof. Consider the cosimplicial vector space V � with

V n D �.NnUI ��DR. xC �

.JX /Œ1�/!/

and the cosimplicial structure induced by the simplicial structure of NU. The coho-mology of the complex .V �; @/ is the Cech cohomology of U with the coefficients inthe soft sheaf of vector spaces �� ˝ xC �.JX /Œ1� and, therefore, vanishes in the posi-tive degrees. GDR.J/! as a cosimplicial vector space can be identified with yV � in thenotations of Lemma 2.1. Hence the result follows from Lemma 2.1.

We leave the proof of the following lemma to the reader.

Lemma 7.9. The map �� W gDR.JX /! ! GDR.J/0! induces an isomorphism of DGLA

gDR.JX /! Š ker.GDR.J/0! � GDR.J/

1!/

where the two maps on the right are .@00/� and .@01/�.

Two previous lemmas together with Corollary 3.13 imply the following:

Proposition 7.10. Let ! 2 �.X I .�2X ˝ JX=OX /cl/ and let m be a commutative

nilpotent ring. Then the map �� induces equivalence of groupoids:

MC2.gDR.JX /! ˝m/ Š Stack.GDR.J/! ˝m/:


For ˇ 2 �.X I�1 ˝ JX=OX / there is a canonical isomorphism of cosimplicialDGLA exp.�ˇ / W GDR.J/!Crcanˇ ! GDR.J/! . Therefore, GDR.J/! depends only onthe class of ! in H 2

DR.JX=OX /.The rest of this section is devoted to the proof of the following theorem.

Theorem 7.11. Suppose that .U;A/ is a descent datum representing a twisted formS of OX . There exists a quasi-isomorphism of cosimplicial DGLA GDR.J/ŒS� !GDR.J.A//.

7.4 Quasiisomorphism. Suppose that .U;A/ is a descent datum for a twisted formof OX . Thus, A is identified with ON0U and A01 is a line bundle on N1U.

7.4.1 Multiplicative connections. Let C�.A01/ denote the set of connections r onA01 which satisfy

1. Ad A012..pr202/�r/ D .pr201/

�r ˝ IdC Id˝ .pr212/�r,

2. .pr000/�r is the canonical flat connection on ON0U.

Let Isom�0 .A01˝JN1U;J.A01//denote the subset of Isom0.A01˝JN1U;J.A01//

which consists of � which satisfy

1. Ad A012..pr202/��/ D .pr201/

�� ˝ .pr212/�� ,

2. .pr000/�� D Id

Note that the vector space xZ1.UI�1/ of cocycles in the normalized Cech complexof the cover U with coefficients in the sheaf of 1-forms �1 acts on the set C�.A01/,with the action given by

˛ � r D r C ˛: (7.3)

Here r 2 C�.A01/, ˛ 2 xZ1.UI�1/ �1.N1U/.Similarly, the vector space xZ1.UIJ0/ acts on the set Isom�

0 .A01˝JN1U;J.A01//,with the action given as in Corollary 7.4.

Note that since the sheaves involved are soft, cocycles coincide with coboundaries:xZ1.UI�1/ D xB1.UI�1/, xZ1.UIJ0/ D xB1.UIJ0/.Proposition 7.12. The set C�.A01/ (respectively, Isom�

0 .A01 ˝ JN1U;J.A01//)is an affine space with the underlying vector space being xZ1.UI�1/ (respectively,xZ1.UIJ0/).Proof. Proofs of the both statements are completely analogous. Therefore we explainthe proof of the statement concerning Isom�

0 .A01 ˝ JN1U;J.A01// only.We show first that Isom�

0 .A01 ˝ JN1U;J.A01// is nonempty. Choose an arbi-trary � 2 Isom0.A01 ˝ JN1U;J.Aij // such that .pr000/

�� D Id. Then, by Corol-lary 7.4, there exists c 2 �.N2UIJ0/ such that c �.Ad A012..d

1/��02// D .d0/��01˝.d2/��12. It is easy to see that c 2 xZ2.UI exp J0/. Since the sheaf exp J0 is soft,


corresponding Cech cohomology is trivial. Therefore, there exists 2 xC 1.UIJ0/such that c D L@. Then, � � 2 Isom�

0 .A01 ˝ JN1U;J.A01//.Suppose that �; � 0 2 Isom�

0 .A01 ˝ JN1U;J.A01//. By Corollary 7.4 � D � � 0for some uniquely defined 2 �.N2UIJ0/. It is easy to see that 2 xZ1.UIJ0/.

We assume from now on that we have chosen � 2 Isom�0 .A01 ˝ JN1U;J.A01//,

r 2 C�.A01/. Such a choice defines �pij 2 Isom0.Aij ˝ JNpU;J.Aij // for every pand 0 � i; j � p by �pij D .prpij /

�� . This collection of �pij induces for every p algebraisomorphism �p W Mat.A/p ˝ JNpU ! Mat.J.A//p . The following compatibilityholds for these isomorphisms. Let f W Œp� ! Œq� be a morphism in �. Then thefollowing diagram commutes:

f�.Mat.A/p ˝ JNpU/f� ��

f�.�p/

��

f ]Mat.A˝ JNqU/q

f ].�p/

��

f�Mat.J.A//pf� �� f ]Mat.J.A//q .

(7.4)

Similarly define the connections rpij D .prpij /�r. For p D 0; 1; : : : set rp D

˚pi;jD0rij ; the connections rp on Mat.A/p satisfy

f�rp D .Ad f�/.f ]rq/: (7.5)

Note that F.�;r/ 2 �.N1UI�1N1U˝JN1U/ is a cocycle of degree 1 in LC �.UI�1X ˝

JX /. Vanishing of the corresponding Cech cohomology implies that there exists F 0 2�.N0UI�1N0U

˝ JN0U/ such that

.d1/�F 0 � .d0/�F 0 D F.�;r/: (7.6)

For p D 0; 1; : : :, 0 � i � p, let F pii D .prpi /�F 0; put F pij D 0 for i ¤ j .

Let F p 2 �.NpUI�1NpU

˝ Mat.A/p/ ˝ JNpU denote the diagonal matrix with

components F pij . For f W Œp�! Œq� we have

f�F p D f ]F q : (7.7)

Then, we obtain the following equality of connections on Mat.A/p ˝ JNpU:

.�p/�1 ı rcan ı �p D rp ˝ IdC Id˝rcan C adF p: (7.8)

The matrices F p also have the following property. Let rcanF p be the diagonalmatrix with the entries .rcanF p/i i D rcanF

pii . Denote byrcanF p the image ofrcanF p

under the natural map �.NpUI�2NpU

˝ Mat.A/p ˝ JNpU/ ! �.NpUI�2NpU

˝Mat.A/p ˝ .JNpU=ONpU//. Recall the canonical map �p W NpU ! X . Then, wehave the following:


Lemma 7.13. There exists a unique ! 2 �.X I .�2X ˝ JX=OX /cl/ such that

rcanF p D ��p! ˝ Idp;

where Idp denotes the .p C 1/ � .p C 1/ identity matrix.

Proof. Using the definition of F 0 and formula (7.2) we obtain: .d1/�rcanF 0 �.d0/�rcanF 0 D rcanF.�;r/2�2.N1U/. Hence .d1/�rcanF 0�.d0/�rcanF 0 D 0,and there exists a unique ! 2 �.X I�2X ˝ .JX=OX // such that ��0! D rcanF 0.Since .rcan/2 D 0 it follows that rcan! D 0. For any p we have: .rcanF p/i i Dpr�i rcanF 0 D ��p!, and the assertion of the lemma follows.

Lemma 7.14. The class of ! in H 2.�.X I��

X ˝ JX=OX /;rcan/ does not depend onthe choices made in its construction.

Proof. The construction of ! is dependent on the choice of F 0, � 2 Isom�0 .A01 ˝

JN1UJ.A01//, and r 2 C�.A01/ satisfying the equation (7.6). Assume that wemake different choices: � 0 D .L@/ � � , r 0 D .L@˛/ � r and .F 0/0 satisfying [email protected] 0/0 DF.� 0;r 0/. Here, 2 xC 0.UIJ0/ and ˛ 2 xC 0.UI�1/. We have: F.� 0;r 0/ DF.�;r/ � L@˛ C L@rcan. It follows that [email protected] 0/0 � F 0 � rcan C ˛/ D 0. Therefore.F 0/0 � F 0 � rcan C ˛ D ��0ˇ for some ˇ 2 �.X I�1X ˝ JX /. Hence if !0 isconstructed using � 0, r 0, .F 0/0 then !0 �! D rcan N where N is the image of ˇ underthe natural projection �.X I�1X ˝ JX /! �.X I�1X ˝ .JX=OX //.

Let � W V ! U be a refinement of the cover U, and .V ;A�/ the correspondingdescent datum. Choice of � , r, F 0 on U induces the corresponding choice .N�/�� ,.N�/�r, .N�/�.F 0/ on V . Let !� denotes the form constructed as in Lemma 7.13using .N�/�� , .N�/�r, .N�/�F 0. Then,

!� D .N�/�!:The following result now follows easily and we leave the details to the reader.

Proposition 7.15. The class of ! in H 2.�.X I��

X ˝ JX=OX /;rcan/ coincides withthe image ŒS � of the class of the gerbe.

7.5 Construction of the quasiisomorphism. For � W Œn�! � let

H� WD �.N�.n/UI ��

N�.n/U˝ �.0n/�C �

.Mat.A/�.0/ ˝ JN�.0/U/locŒ1�/;

considered as a graded Lie algebra. For W Œm�! Œn�, � D � ı there is a morphismof graded Lie algebras � W H� ! H�. For n D 0; 1; : : : let Hn WD Q

Œn��!�

H�. The

assignment � 3 Œn� 7! Hn, 7! � defines a cosimplicial graded Lie algebra H.For each � W Œn�! � the map

�� WD Id˝ �.0n/�.��.0// W H� ! GDR.J.A//�


is an isomorphism of graded Lie algebras. It follows from (7.4) that the maps �� yieldan isomorphism of cosimplicial graded Lie algebras

�� W H! GDR.J.A//:

Moreover, the equation (7.8) shows that if we equip H with the differential given on�.N�.n/UI ��

N�.n/U˝ �.0n/�C �.Mat.A/�.0/ ˝ JN�.0/U/

locŒ1�/ by

ı C �.0n/�.r�.0//˝ IdC Id˝rcan C ad �.0n/�.F �.0//D ı C �.0n/].r�.n//˝ IdC Id˝rcan C ad �.0n/].F �.n/// (7.9)

then �� becomes an isomorphism of DGLA. Consider now an automorphism exp �F ofthe cosimplicial graded Lie algebra H given on H� by exp ��.0n/�Fp . Note the fact thatthis morphism preserves the cosimplicial structure follows from the relation (7.7).

The following result is proved by the direct calculation; see [4], Lemma 16.

Lemma 7.16.

exp.�Fp / ı .ı Crp ˝ IdC Id˝rcan C adF p/ ı exp.��Fp /D ı Crp ˝ IdC Id˝rcan � �rF p : (7.10)

Therefore the morphismexp �F W H! H (7.11)

conjugates the differential given by the formula (7.9) into the differential which on H�

is given by

ı C �.0n/�.r�.0//˝ IdC Id˝rcan � ��.0n/�.rcanF �.0//: (7.12)

Consider the map

cotr W xC �

.JNpU/Œ1�! C�

.Mat.A/p ˝ JNpU/Œ1� (7.13)

defined as follows:

cotr.D/.a1 ˝ j1; : : : ; an ˝ jn/ D a0 : : : anD.j1; : : : ; jn/: (7.14)

The map cotr is a quasiisomorphism of DGLAs (cf. [15], section 1.5.6; see also [4]Proposition 14).

Lemma 7.17. For every p the map

Id˝ cotr W ��

NpU ˝ xC �

.JNpU/Œ1�! ��

NpU ˝ C �

.Mat.A/p ˝ JNpU/Œ1� (7.15)

is a quasiisomorphism of DGLA, where the source and the target are equipped with thedifferentials ı Crcan C ��! and ı Crp ˝ IdC Id˝rcan � �rF p respectively.


Proof. It is easy to see that Id˝cotr is a morphism of graded Lie algebras, which satisfies.rp˝IdCId˝rcan/ı.Id˝cotr/ D .Id˝cotr/ırcan and ıı.Id˝cotr/ D .Id˝cotr/ıı.Since the domain of .Id˝ cotr/ is the normalized complex, in view of Lemma 7.13 wealso have �rF p ı .Id˝ cotr/ D �.Id˝ cotr/ ı ��! . This implies that .Id˝ cotr/ is amorphism of DGLA.

To see that this map is a quasiisomorphism, introduce filtration on ��

NpUby

Fi��

NpUD �

��iNpU

and consider the complexes xC �.JNpU/Œ1� and C �.Mat.A//p ˝JNpU/Œ1� equipped with the trivial filtration. The map (7.15) is a morphism of filteredcomplexes with respect to the induced filtrations on the source and the target. Thedifferentials induced on the associated graded complexes are ı (or, more precisely,Id ˝ ı) and the induced map of the associated graded objects is Id ˝ cotr which is aquasi-isomorphism. Therefore, the map (7.15) is a quasiisomorphism as claimed.

The map (7.15) therefore induces a morphism Id˝cotr W GDR.J/�! ! H� for every

� W Œn�! �. These morphisms are clearly compatible with the cosimplicial structureand hence induce a quasiisomorphism of cosimplicial DGLAs

Id˝ cotr W GDR.J/! ! H

where the differential in the right hand side is given by (7.12).We summarize our consideration in the following:

Theorem 7.18. For a any choice of � 2 Isom�0 .A01˝JN1U;J.A01//,r 2 C�.A01/

and F 0 as in (7.6) the composition ˆ�;r;F WD �� ı exp.�F / ı .Id˝ cotr/

ˆ�;r;F W GDR.J/! ! GDR.J.A// (7.16)

is a quasiisomorphism of cosimplicial DGLAs.

Let� W V ! U be a refinement of the cover U and let .V ;A�/be the induced descentdatum. We will denote the corresponding cosimplicial DGLAs by GDR.J/!.U/ andGDR.J/!.V/ respectively. Then the map N� W NV ! NU induces a morphism ofcosimplicial DGLAs

.N�/� W GDR.J.A//! GDR.J.A�// (7.17)

and.N�/� W GDR.J/!.U/! GDR.J/!.V/: (7.18)

Notice also that the choice that the choice of data � , r, F 0 on NU induces thecorresponding data .N�/�� , .N�/�r, .N�/�F 0 on NV . This data allows one toconstruct using the equation (7.16) the map

ˆ.N�/��;.N�/�r;.N�/�F W GDR.J/! ! GDR.J.A�//: (7.19)

The following proposition is an easy consequence of the description of the map ˆ,and we leave the proof to the reader.


Proposition 7.19. The following diagram commutes:

GDR.J/!.U/.N�/�

��

ˆ�;r;F

��

GDR.J/!.V/

ˆ.N�/��;.N�/�r;.N�/�F

��

GDR.J.A//.N�/�

�� GDR.J.A�//.

(7.20)

8 Proof of the main theorem

In this section we prove the main result of this paper. Recall the statement of thetheorem from the introduction:

Theorem 1. Suppose thatX is aC1 manifold and S is an algebroid stack onX whichis a twisted form of OX . Then, there is an equivalence of 2-groupoid valued functorsof commutative Artin C-algebras

DefX .S/ Š MC2.gDR.JX /ŒS�/:

Proof. Suppose U is a cover of X such that ��0S.N0U/ is nonempty. There is adescent datum .U;A/ 2 DescC.U/ whose image under the functor DescC.U/ !AlgStackC.X/ is equivalent to S .

The proof proceeds as follows.Recall the 2-groupoids Def 0.U;A/.R/ and Def.U;A/.R/ of deformations of and

star-products on the descent datum .U;A/ defined in Section 6.2. Note that the compo-sition DescR.U/! TrivR.X/! AlgStackR.X/ induces functors Def 0.U;A/.R/!Def.S/.R/ and Def.U;A/.R/! Def.S/.R/, the second one being the compositionof the first one with the equivalence Def.U;A/.R/! Def 0.U;A/.R/. We are goingto show that for a commutative Artin C-algebra R there are equivalences

1. Def.U;A/.R/ Š MC2.gDR.JX /ŒS�/.R/ and

2. the functor Def.U;A/.R/ Š Def.S/.R/ induced by the functor Def.U;A/.R/!Def.S/.R/ above.

Let J.A/ D .J.A01/; j1.A012//. Then, .U;J.A// is a descent datum for a

twisted form of JX . Let R be a commutative Artin C-algebra with maximal ideal mR.Then the first statement follows from the equivalences

Def.U;A/.R/ Š Stack.G.A/˝mR/ (8.1)

Š Stack.GDR.J.A///˝mR/ (8.2)

Š Stack.GDR.JX /ŒS� ˝mR/ (8.3)

Š MC2.gDR.JX /ŒS� ˝mR/: (8.4)

Here the equivalence (8.1) is the subject of Proposition 6.4. The inclusion of hor-izontal sections is a quasi-isomorphism and the induced map in (8.2) is an equiva-lence by Theorem 3.6. In Theorem 7.18 we have constructed a quasi-isomorphism


G.JX /ŒS� ! G.J.A//; the induced map (8.3) is an equivalence by another applica-tion of Theorem 3.6. Finally, the equivalence (8.4) is shown in Proposition 7.10

We now prove the second statement. We begin by considering the behavior ofDef 0.U;A/.R/ under the refinement. Consider a refinement � W V ! U . Recallthat by Proposition 7.10 the map �� induces equivalences MC2.gDR.JX /! ˝m/ !Stack.GDR.J/!.U/˝m/ and MC2.gDR.JX /! ˝m/ ! Stack.GDR.J/!.V/˝m/.It is clear that the diagram

MC2.gDR.JX /! ˝m/

��

��

Stack.GDR.J/!.U/˝m/.N�/�

�� Stack.GDR.J/!.V/˝m/

commutes, and therefore .N�/� W Stack.GDR.J/!.U/˝m/! Stack.GDR.J/!.V/˝m/ is an equivalence. Then Proposition 7.19 together with Theorem 3.6 implies that.N�/� W Stack.GDR.J.A/// ˝ m/ ! Stack.GDR.J.A

�/// ˝ m/ is an equivalence.It follows that the functor �� W Def.U;A/.R/ ! Def.V ;A�/.R/ is an equivalence.Note also that the diagram

Def.U;A/.R/

��

��

�� Def.V ;A�/.R/

��

Def 0.U;A/.R/ ��

�� Def 0.V ;A�/.R/

is commutative with the top horizontal and both vertical maps being equivalences.Hence it follows that the bottom horizontal map is an equivalence.

Recall now that the embedding Def.U;A/.R/! Def 0.U;A/.R/ is an equivalenceby Proposition 6.3. Therefore it is sufficient to show that the functor Def 0.U;A/.R/!Def.S/.R/ is an equivalence. Suppose that C is anR-deformation of S . It follows fromLemma 6.2 that C is an algebroid stack. Therefore, there exists a cover V and an R-descent datum .V ;B/ whose image under the functor DescR.V/! AlgStackR.X/ isequivalent to C . Replacing V by a common refinement of U and V if necessary we mayassume that there is a morphism of covers � W V ! U. Clearly, .V ;B/ is a deformationof .V ;A�/. Since the functor �� W Def 0.U;A/.R/! Def 0.V ;A�/ is an equivalencethere exists a deformation .U;B 0/ such that ��.U;B 0/ is isomorphic to .V ;B/. LetC 0 denote the image of .U;B 0/ under the functor DescR.V/! AlgStackR.X/. Sincethe images of .U;B 0/ and ��.U;B 0/ in AlgStackR.X/ are equivalent it follows thatC 0 is equivalent to C . This shows that the functor Def 0.U;A/.R/ ! Def.C/.R/ isessentially surjective.

Suppose now that .U;B.i//, i D 1; 2, are R-deformations of .U;A/. Let C .i/

denote the image of .U;B.i// in Def.S/.R/. Suppose that F W C .1/ ! C .2/ is a1-morphism.


Let L denote the image of the canonical trivialization under the composition

��0 .F / W AB.1/C Š ��0C .1/F�! ��0C .1/ Š A

B.2/C:

Thus, L is a B.1/ ˝R B.2/op-module such that the line bundle L ˝R C is trivial.Therefore, L admits a non-vanishing global section. Moreover, there is an isomor-phism f W B.1/

01 ˝.B.1//11.pr11/

�L! .pr10/�L˝.B.2//1

0B.2/01 of .B.1//10˝R ..B.2//11/

op-modules.

A choice of a non-vanishing global section ofL gives rise to isomorphisms B.1/01 Š

B.1/01 ˝.B.1//1

1.pr11/

�L and B.2/01 Š .pr10/

�L˝.B.2//10

B.2/01 .

The composition

B.1/01 Š B

.1/01 ˝.B.1//1

1.pr11/

�Lf�! .pr10/

�L˝.B.2//10

B.2/01 Š B

.2/01

defines a 1-morphism of deformations of .U;A/ such that the induced 1-morphismC .1/ ! C .2/ is isomorphic to F . This shows that the functor Def 0.U;A/.R/ !Def.C/ induces essentially surjective functors on groupoids of morphisms. By similararguments left to the reader one shows that these are fully faithful.

This completes the proof of Theorem 1.

References

[1] Revêtements étales et groupe fondamental (SGA 1). Documents Mathématiques (Paris), 3.Société Mathématique de France, Paris, 2003, Séminaire de géométrie algébrique du BoisMarie 1960–61, directed by A. Grothendieck, with two papers by M. Raynaud, updatedand annotated reprint of the 1971 original.

[2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation the-ory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978),61–110.

[3] L. Breen, On the classification of 2-gerbes and 2-stacks, Astérisque 225 (1994).

[4] P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan, Deformations of Azumaya algebras,in Actas del XVI Coloquio Latinoamericano de Algebra (Colonia del Sacramento, Uruguay,Agosto 2005), Biblioteca de la Revista Matemática Iberoamericana, 2008, 131–152.

[5] P. Bressler, A. Gorokhovsky, R.Nest, and B. Tsygan, Deformation quantization of gerbes,Adv. Math. 214 (2007), 230–266.

[6] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progr.Math. 107, Birkhäuser, Boston, MA, 1993.

[7] A. D’Agnolo and P. Polesello, Stacks of twisted modules and integral transforms, in Geo-metric aspects of Dwork theory, Vol. I, Walter de Gruyter, Berlin 2004, 463–507.

[8] J. Dixmier and A. Douady, Champs continus d’espaces hilbertiens et de C�-algèbres, Bull.Soc. Math. France 91 (1963), 227–284.

[9] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2), 79 (1964),59–103.


[10] E. Getzler, Lie theory for nilpotent L1-algebras, Ann. of Math., to appear, preprint 2007,arXiv:math.AT/0404003.

[11] E. Getzler, A Darboux theorem for Hamiltonian operators in the formal calculus of varia-tions, Duke Math. J. 111 (2002), 535–560.

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[13] M. Kontsevich, Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001),271–294.

[14] M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66(2003), 157–2160.

[15] J.-L. Loday, Cyclic homology, second edition, Grundlehren Math. Wiss. 301, Springer-Verlag, Berlin 1998.

[16] W. Lowen, Algebroid prestacks and deformations of ringed spaces, Trans. Amer. Math. Soc.360 (2008), 1631–1660

[17] W. Lowen and M. Van den Bergh, Hochschild cohomology of abelian categories and ringedspaces, Adv. Math. 198 (2005), 172–221.

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[19] M. Schlessinger and J. Stasheff, The Lie algebra structure of tangent cohomology anddeformation theory, J. Pure Appl. Algebra 38 (1985), 313–322.

[20] A. Vistoli, Grothendieck topologies, fibered categories and descent theory, in Fundamentalalgebraic geometry, Math. Surveys Monogr. 123, Amer. Math. Soc., Providence, RI, 2005,1–104.

Torsion classes of finite type and spectra

Grigory Garkusha� and Mike Prest

1 Introduction

Non-commutative geometry comes in various flavours. One is based on abelian and tri-angulated categories, the latter being replacements of classical schemes. This is basedon classical results of Gabriel and later extensions, in particular byThomason. Precisely,Gabriel [6] proved that any noetherian scheme X can be reconstructed uniquely up toisomorphism from the abelian category, QcohX , of quasi-coherent sheaves over X .This reconstruction result has been generalized to quasi-compact schemes by Rosen-berg in [15]. Based on Thomason’s classification theorem, Balmer [3] reconstructs anoetherian scheme X from the triangulated category of perfect complexes Dper.X/.This result has been generalized to quasi-compact, quasi-separated schemes by Buan–Krause–Solberg [5].

In this paper we reconstruct affine and projective schemes from appropriate abeliancategories. Our approach, similar to that used in [8], [9], is different from Rosen-berg’s [15] and less abstract. Moreover, some results of the paper are of independentinterest.

Let ModR (respectively QGrA) denote the category of R-modules (respectivelygraded A-modules modulo torsion modules) with R (respectively A D L

n�0An) acommutative ring (respectively a commutative graded ring). We first demonstrate thefollowing result (cf. [8], [9]).

Theorem (Classification). Let R (respectively A) be a commutative ring (respectivelycommutative graded ring which is finitely generated as anA0-algebra). Then the maps

V 7! S D fM 2 ModR j suppR.M/ � V g; S 7! V D[M2S

suppR.M/

and

V 7! S D fM 2 QGrA j suppA.M/ � V g; S 7! V D[M2S

suppA.M/

induce bijections between

1. the set of all subsets V � SpecR (respectively V � ProjA) of the form V DSi2� Yi with SpecR nYi (respectively ProjA nYi ) quasi-compact and open for

all i 2 �,�This paper was written during the visit of the first author to the University of Manchester supported by

the MODNET Research Training Network in Model Theory. He would like to thank the University for thekind hospitality.

394 G. Garkusha and M. Prest

2. the set of all torsion classes of finite type in ModR (respectively tensor torsionclasses of finite type in QGrA).

This theorem says that SpecR and ProjA contain all the information about finitelocalizations in ModR and QGrA respectively. The next result says that there isa 1-1 correspondence between the finite localizations in ModR and the triangulatedlocalizations in Dper.R/ (cf. [11], [8]).

Theorem. Let R be a commutative ring. The map

S 7! T D fX 2 Dper.R/ j Hn.X/ 2 S for all n 2 Zginduces a bijection between

1. the set of all torsion classes of finite type in ModR,

2. the set of all thick subcategories of Dper.R/.

Following Buan–Krause–Solberg [5] we consider the lattices Ltor.ModR/ andLtor.QGrA/ of (tensor) torsion classes of finite type in ModR and QGrA, as well astheir prime ideal spectra Spec.ModR/ and Spec.QGrA/. These spaces come naturallyequipped with sheaves of rings OModR and OQGrA. The following result says that theschemes .SpecR;OR/ and .ProjA;OProjA/ are isomorphic to .Spec.ModR/;OModR/

and .Spec.QGrA/;OQGrA/ respectively.

Theorem (Reconstruction). LetR (respectivelyA) be a commutative ring (respectivelycommutative graded ring which is finitely generated as anA0-algebra). Then there arenatural isomorphisms of ringed spaces

.SpecR;OR/��! .Spec.ModR/;OModR/

and

.ProjA;OProjA/��! .Spec.QGrA/;OQGrA/:

2 Torsion classes of finite type

We refer the reader to the appendix for necessary facts about localization and torsionclasses in Grothendieck categories.

Proposition 2.1. Assume that B is a set of finitely generated ideals of a commutativering R. The set of those ideals which contain a finite products of ideals belonging toB is a Gabriel filter of finite type.

Proof. See [17, VI.6.10].

Given a module M , we denote by suppR.M/ D fP 2 SpecR j MP ¤ 0g. HereMP denotes the localization ofM at P , that is, the module of fractionsMŒ.R nP /�1�.

Torsion classes of finite type and spectra 395

Note that V.I / D fP 2 SpecR j I � P g is equal to suppR.R=I / for every ideal Iand

suppR.M/ D[x2M

V.annR.x//; M 2 ModR:

Recall from [10] that a topological space is spectral if it is T0, quasi-compact, ifthe quasi-compact open subsets are closed under finite intersections and form an openbasis, and if every non-empty irreducible closed subset has a generic point. Given aspectral topological space, X , Hochster [10] endows the underlying set with a new,“dual", topology, denoted X�, by taking as open sets those of the form Y D S

i2� Yiwhere Yi has quasi-compact open complementX nYi for all i 2 �. ThenX� is spectraland .X�/� D X (see [10, Proposition 8]). The spaces, X , which we shall consider arenot in general spectral; nevertheless we make the same definition and denote the spaceso obtained by X�.

Given a commutative ring R, every closed subset of SpecR with quasi-compactcomplement has the form V.I / for some finitely generated ideal, I , of R (see [2,Chapter 1, Ex. 17 (vii)]). Therefore a subset of Spec�R is open if and only if it is of theform

S� V.I�/ with each I� finitely generated. Notice that V.I / with I a non-finitely

generated ideal is not open in Spec�R in general. For instance (see [18, 3.16.2]), letR D CŒx1; x2; : : : � and m D .x1; x2; : : : /. It is clear that V.m/ D fmg is not open inSpec�CŒx1; x2; : : : �.

For definitions of terms used in the next result see the Appendix to this paper.

Theorem 2.2 (Classification). Let R be a commutative ring. There are bijectionsbetween

1. the set of all open subsets V � Spec�R,

2. the set of all Gabriel filters F of finite type,

3. the set of all torsion classes S of finite type in ModR.

These bijections are defined as follows:

V 7!´

FV D ¹I � R j V.I / � V º;SV D ¹M 2 ModR j suppR.M/ � V ºI

F 7!

8<:VF D

[I2F

V.I /;

SF D ¹M 2 ModR j annR.x/ 2 F for every x 2M ºI

S 7!

8<:

FS D ¹I � R j R=I 2 Sº;VS D

[M2S

suppR.M/:


Proof. The bijection between Gabriel filters of finite type and torsion classes of finitetype is a consequence of a theorem of Gabriel (see, e.g., [7, 5.8]).

Let F be a Gabriel filter of finite type. Then the setƒF of finitely generated idealsI belonging to F is a filter basis for F. Therefore VF D

SI2ƒF

V.I / is open inSpec�R.

Now let V be an open subset of Spec�R. Letƒ denote the set of finitely generatedideals I such that V.I / � V . By definition of the topology V D S

I2ƒ V.I / andI1 � � � In 2 ƒ for any I1; : : : ; In 2 ƒ. We denote by F0V the set of ideals I � R

such that I � J for some J 2 ƒ. By Proposition 2.1 F0V is a Gabriel filter of finitetype. Clearly, F0V � FV D fI � R j V.I / � V g. Suppose I 2 FV n F0V ; by [17,VI.6.13–15] (cf. the proof of Theorem 6.4) there exists a prime ideal P 2 V.I / suchthat P 62 F0V . But V.I / � V and therefore P � J for some J 2 ƒ, so P 2 F0V , acontradiction. Thus F0V D FV .

Clearly, V D VFV for every open subset V � Spec�R. Let F be a Gabriel filterof finite type and I 2 F. Clearly F � FVF

and, as above, there is no ideal belongingto FVF

nF. We have shown the bijection between the sets of all Gabriel filters of finitetype and all open subsets in Spec�R. The description of the bijection between the setof torsion classes of finite type and the set of open subsets in Spec�R is now easilychecked.

3 The fg-topology

Let InjR denote the set of isomorphism classes of indecomposable injective modules.Given a finitely generated ideal I of R, we denote by SI the torsion class of finitetype corresponding to the Gabriel filter of finite type having fI ngn�1 as a basis (seeProposition 2.1 and Theorem 2.2). Note that a moduleM has SI -torsion if and only ifevery element x 2M is annihilated by some power I n.x/ of the ideal I . Let us set

Dfg.I / WD fE 2 InjR j E is SI -torsion freeg; V fg.I / WD InjR nDfg.I /

(“fg” referring to this topology being defined using only finitely generated ideals).LetE be any indecomposable injectiveR-module. Set P D P.E/ to be the sum of

annihilator ideals of non-zero elements, equivalently non-zero submodules, ofE. SinceE is uniform the set of annihilator ideals of non-zero elements of E is closed underfinite sum. It is easy to check ([14, 9.2]) that P.E/ is a prime ideal and P.EP / D P .Here EP stands for the injective hull of R=P . There is an embedding

˛ W SpecR! InjR; P 7! EP ;

which need not be surjective. We shall identify SpecR with its image in InjR.If P is a prime ideal of a commutative ring R its complement in R is a multiplica-

tively closed set S . Given a module M we denote the module of fractions MŒS�1�by MP . There is a corresponding Gabriel filter

FP D fI j P … V.I /g:


Clearly, FP is of finite type. The FP -torsion modules are characterized by the propertythat MP D 0 (see [17, p. 151]).

More generally, let P be a subset of SpecR. To P we associate a Gabriel filter

FP D\P2P

FP D fI j P \ V.I / D ;g:

The corresponding torsion class consists of all modulesM withMP D 0 for allP 2 P .Given a family of injective R-modules E , denote by FE the Gabriel filter de-

termined by E . By definition, this corresponds to the localizing subcategory SE DfM 2 ModR j HomR.M;E/ D 0 for all E 2 Eg.Proposition 3.1. A Gabriel filter F is of finite type if and only if it is of the form FP

with P a closed set in Spec�R. Moreover, FP is determined by EP D fEP j P 2 P gvia FP D fI j HomR.R=I;EP / D 0g.Proof. This is a consequence of Theorem 2.2.

Proposition 3.2. Let P be the closure of P in Spec�R. Then P D fQ 2 SpecR jQ � P g. Also FP D FP .

Proof. This is direct from the definition of the topology.

Recall that for any ideal I of a ring, R, and r 2 R we have an isomorphismR=.I W r/ Š .rR C I /=I , where .I W r/ D fs 2 R j rs 2 I g, induced by sending1C .I W r/ to r C I .

Proposition 3.3. Let E be an indecomposable injective module and let P.E/ be theprime ideal defined before. Let I be a finitely generated ideal of R. Then E 2 V fg.I /

if and only if EP.E/ 2 V fg.I /.

Proof. Let I be such that E D E.R=I /. For each r 2 R n I we have, by the remarkjust above, that the annihilator of rCI 2 E is .I W r/ and so, by definition ofP.E/;wehave .I W r/ � P.E/. The natural projection .rR C I /=I Š R=.I W r/ ! R=P.E/

extends to a morphism from E to EP.E/ which is non-zero on r C I . Forming theproduct of these morphisms as r varies over R n I , we obtain a morphism from E toa product of copies of EP.E/ which is monic on R=I and hence is monic. ThereforeE is a direct summand of a product of copies of EP.E/ and so E 2 V fg.J / impliesEP.E/ 2 V fg.J /, where J is a finitely generated ideal.

Now, EP.E/ 2 V fg.I /, where I is a finitely generated ideal, means that there isa non-zero morphism f W R=I n ! EP.E/ for some n. Since R=P.E/ is essential inEP.E/ the image of f has non-zero intersection with R=P.E/ so there is an ideal J ,without loss of generality finitely generated, with I n < J � R, J=I n a cyclic module,and such that the restriction, f 0, of f to J=I n is non-zero (and the image is containedin R=P.E/). Since J=I n is a cyclic SI -torsion module, there is an epimorphismg W R=Im ! J=I n for some m. By construction, R=P.E/ D lim�!R=I�, where I�


ranges over the annihilators of non-zero elements of E. Since R=Im is finitely pre-sented, 0 ¤ f 0g factorises through one of the maps R=I� ! R=P.E/. In particular,there is a non-zero morphismR=Im ! E showing thatE 2 V fg.I /, as required.

Given a module M , we set

ŒM � WD fE 2 InjR j HomR.M;E/ D 0g; .M/ WD InjR n ŒM �:

Remark 3.4. For any finitely generated ideal I we have: Dfg.I / \ SpecR D D.I/

and V fg.I / \ SpecR D V.I /. Moreover, Dfg.I / D ŒR=I �.If I; J are finitely generated ideals, then D.IJ / D D.I/ \D.J /. It follows from

Proposition 3.3 and Remark 3.4 that Dfg.I / \ Dfg.J / D Dfg.IJ /. Thus the setsDfg.I / with I running over finitely generated ideals form a basis for a topology onInjR which we call the fg-ideals topology. This topological space will be denoted byInjfgR. Observe that if R is coherent then the fg-topology equals the Zariski topologyon InjR (see [14], [8]). The latter topological space is defined by taking the ŒM � withM finitely presented as a basis of open sets.

Theorem 3.5. (cf. Prest [14, 9.6]) Let R be a commutative ring, let E be an indecom-posable injective module and let P.E/ be the prime ideal defined before. Then E andEP.E/ are topologically indistinguishable in InjfgR.

Proof. This follows from Proposition 3.3 and Remark 3.4.

Theorem 3.6. (cf. Garkusha–Prest [8, Theorem A]) Let R be a commutative ring.The space SpecR is dense and a retract in InjfgR. A left inverse to the embeddingSpecR ,! InjfgR takes an indecomposable injective module E to the prime idealP.E/. Moreover, InjfgR is quasi-compact, the basic open subsetsDfg.I /, with I finitelygenerated, are quasi-compact, the intersection of two quasi-compact open subsets isquasi-compact, and every non-empty irreducible closed subset has a generic point.

Proof. For any finitely generated ideal I we have

Dfg.I / \ SpecR D D.I/(see Remark 3.4). From this relation and Theorem 3.5 it follows that SpecR is densein InjfgR and that ˛ W SpecR! InjfgR is a continuous map.

One may check (see [14, 9.2]) that

ˇ W InjfgR! SpecR; E 7! P.E/;

is left inverse to ˛. Remark 3.4 implies that ˇ is continuous. Thus SpecR is a retractof InjfgR.

Let us show that each basic open setDfg.I / is quasi-compact (in particular InjfgR DDfg.R/ is quasi-compact). LetDfg.I / DS

i2�Dfg.Ii /with each Ii finitely generated.It follows from Remark 3.4 that D.I/ D S

i2�D.Ii /. Since I is finitely generated,


D.I/ is quasi-compact in SpecR by [2, Chapter 1, Ex. 17 (vii)]. We see that D.I/ DSi2�0 D.Ii / for some finite subset �0 � �.Assume E 2 Dfg.I / nS

i2�0 Dfg.Ii /. It follows from Theorem 3.5 that EP.E/ 2

Dfg.I /nSi2�0 Dfg.Ii /. ButEP.E/ 2 Dfg.I /\SpecR D D.I/ DS

i2�0 D.Ii /, andhence it is inD.Ii0/ D Dfg.Ii0/\SpecR for some i0 2 �0, a contradiction. SoDfg.I /

is quasi-compact. It also follows that the intersection Dfg.I /\Dfg.J / D Dfg.IJ / oftwo quasi-compact open subsets is quasi-compact. Furthermore, every quasi-compactopen subset in InjfgR must therefore have the form Dfg.I / with I finitely generated.

Finally, it follows from Remark 3.4 and Theorem 3.5 that a subset V of InjfgR isZariski-closed and irreducible if and only if there is a prime ideal Q of R such thatV D fE j P.E/ Qg. This obviously implies that the pointEQ 2 V is generic.

Notice that InjfgR is not a spectral space in general, for it is not necessarily T0.

Lemma 3.7. Let the ring R be commutative. Then the maps

Spec�R � V �7! QV D fE 2 InjR j P.E/ 2 V gand

.InjfgR/� � Q

7! VQ D fP.E/ 2 Spec�R j E 2 Qg D Q \ Spec�R

induce a 1-1 correspondence between the lattices of open sets of Spec�R and those of.InjfgR/�.

Proof. First note that EP 2 QV for any P 2 V (see [14, 9.2]). Let us check thatQV is an open set in .InjfgR/�. Every closed subset of SpecR with quasi-compactcomplement has the form V.I / for some finitely generated ideal, I , of R (see [2,Chapter 1, Exercise 17 (vii)]), so there are finitely generated ideals I� � R such thatV DS

� V.I�/. Since the points E and EP.E/ are, by Theorem 3.5, indistinguishablein .InjfgR/� we see that QV DS

� Vfg.I�/, hence this set is open in .InjfgR/�.

The same arguments imply that VQ is open in Spec�R. It is now easy to see thatVQV D V and QVQ

D Q.

4 Torsion classes and thick subcategories

We shall write L.Spec�R/, L..InjfgR/�/, Lthick.Dper.R//, Ltor.ModR/ to denote:

• the lattice of all open subsets of Spec�R,

• the lattice of all open subsets of .InjfgR/�,

• the lattice of all thick subcategories of Dper.R/,

• the lattice of all torsion classes of finite type in ModR, ordered by inclusion.


(A thick subcategory is a triangulated subcategory closed under direct summands).Given a perfect complex X 2 Dper.R/ denote by supp.X/ D fP 2 SpecR j

X ˝LR RP ¤ 0g. It is easy to see that

supp.X/ D[n2Z

suppR.Hn.X//;

where Hn.X/ is the nth homology group of X .

Theorem 4.1 (Thomason [18]). Let R be a commutative ring. The assignments

T 2 Lthick.Dper.R//�7!

[X2T

supp.X/

andV 2 L.Spec�R/ �7! fX 2 Dper.R/ j supp.X/ � V g

are mutually inverse lattice isomorphisms.

Given a subcategory X in ModR, we may consider the smallest torsion class offinite type in ModR containing X. This torsion class we denote by

pX D

\fS � ModR j S � X is a torsion class of finite typeg:

Theorem 4.2. (cf. Garkusha–Prest [8, Theorem C]) Let R be a commutative ring.There are bijections between

• the set of all open subsets Y � .InjfgR/�,

• the set of all torsion classes of finite type in ModR,

• the set of all thick subcategories of Dper.R/.

These bijections are defined as follows:

Y 7!´

S D ¹M j .M/ � Y º;T D ¹X 2 Dper.R/ j .Hn.X// � Y for all n 2 ZºI

S 7!

8<:Y D

[M2S

.M/;

T D ¹X 2 Dper.R/ j Hn.X/ 2 S for all n 2 ZºI

T 7!

8<:Y D

[X2T ;n2Z

.Hn.X//;

S D p¹Hn.X/ j X 2 T ; n 2 Zº:


Proof. That SY D fM j .M/ � Y g is a torsion class follows because it is defined asthe class of modules having no non-zero morphism to a family of injective modules,E WD InjR n Y . By Lemma 3.7, E \ Spec�R D U is a closed set in Spec�R, that isP.E/ 2 U for all E 2 E . SY is also determined by the family of injective modulesfEP gP2U . Indeed, anyE 2 E is a direct summand of some power ofEP.E/ by the proofof Proposition 3.3. Therefore HomR.M;EP.E// D 0 implies HomR.M;E/ D 0. ByProposition 3.1 SY is of finite type. Conversely, given a torsion class of finite type S , theset YS D

SM2S .M/ is plainly open in .InjfgR/�. Moreover, SYS

D S and Y D YSY .Consider the following diagram:

L.Spec�R/� ��

�

��

Lthick.Dper.R//�

��

�

��

L..InjfgR/�/�

��

��

Ltor.ModR/;ı

��

��

where �; are as in Lemma 3.7,�; � are as in Theorem 4.1 and the remaining maps arethe corresponding maps indicated in the formulation of the theorem. We have � D ��1by Theorem 4.1, � D �1 by Lemma 3.7, and � D ı�1 by the above.

By construction,

��.V / D ˚X jSn2Z suppR.Hn.X// � V

� D fX j supp.X/ � V gfor all V 2 L.Spec�R/. Thus �� D �. Since �, �, � are bijections so is � .

On the other hand,

ı.T / D[

X2T ;n2Z

suppR.Hn.X// D[X2T

supp.X/

for any T 2 Lthick.Dper.R//. We have used here the relation

[M2�.T /

suppR.M/ D[

X2T ;n2Z

suppR.Hn.X//:

One sees that ı D �. Since ı; ; � are bijections so is . Obviously, � D �1and the diagram above yields the desired bijective correspondences. The theorem isproved.

To conclude this section, we should mention the relation between torsion classesof finite type in ModR and the Ziegler subspace topology on InjR (we denote thisspace by InjzgR). The latter topology arises from Ziegler’s work on the model theoryof modules [20]. The points of the Ziegler spectrum of R are the isomorphism classesof indecomposable pure-injective R-modules and the closed subsets correspond tocomplete theories of modules. It is well known (see [14, 9.12]) that for every coherentring R there is a 1-1 correspondence between the open (equivalently closed) subsets


of InjzgR and torsion classes of finite type in ModR. However, this is not the case forgeneral commutative rings.

The topology on InjzgR can be defined as follows. Let M be the set of those modulesM which are kernels of homomorphisms between finitely presented modules; that is

M D Ker.Kf�! L/ with K;L finitely presented. The sets .M/ with M 2 M form

a basis of open sets for InjzgR. We claim that there is a ring R and a module M 2M

such that the intersection .M/\ Spec�R is not open in Spec�R, and hence such thatthe open subset .M/ cannot correspond to any torsion class of finite type on ModR.Such a ring has been pointed out by G. Puninski.

Let V be a commutative valuation domain with value group isomorphic to DLn2Z Z, a Z-indexed direct sum of copies of Z. The order on is defined as follows.

.an/n2Z > .bn/n2Z if ai > bi for some i and ak D bk for every k < i . Then J 2 D Jwhere J is the Jacobson radical of V . Let r be an element with value v.r/ D .an/n2Z

where a0 D 1 and an D 0 for all n 6D 0. Consider the ring R D V=rJ . AgainJ.R/2 D J.R/. Denoting the image of r in R by r 0, note that annR.r 0/ D J.R/

which is not finitely generated, and so R is not coherent by Chase’s Theorem (see [17,1.13.3]). Note thatR is a local ring and, as already observed, the simple moduleR=J.R/is isomorphic to r 0R. Therefore R=J.R/ D Ker.R ! R=r 0R/. Thus R=J.R/ 2M.We have

.R=J.R// \ Spec�R D V.J.R// D fJ.R/g:SupposeV.J.R// is open in Spec�R; thenV.J.R// DS

� V.I�/with each I� finitelygenerated. Since J.R/ is the largest proper ideal each V.I�/, if non-empty, equalsfJ.R/g. Therefore J.R/ D pI� for some �. But the prime radical of every finitelygenerated ideal inR is prime (sinceR is a valulation ring) and different from J.R/. Tosee the latter, we have, since I� is finitely generated, that all elements of I� have value> .a0n/n for some .a0n/n with a0n D 0 for all n � N for some fixed N . (Recall thatthe valuation v on R satisfies v.r C s/ minfv.r/; v.s/g and v.rs/ D v.r/C v.s/.)It follows that there is a prime ideal properly between I� and J.R/. This gives acontradiction, as required.

5 Graded rings and modules

In this section we recall some basic facts about graded rings and modules.

Definition. A (positively) graded ring is a ring A together with a direct sum decom-position A D A0 ˚ A1 ˚ A2 ˚ � � � as abelian groups, such that AiAj � AiCj fori; j 0. A homogeneous element of A is simply an element of one of the groups Aj ,and a homogeneous ideal of A is an ideal that is generated by homogeneous elements.A graded A-module is an A-module M together with a direct sum decompositionM D L

j2ZMj as abelian groups, such that AiMj � MiCj for i 0; j 2 Z. Onecalls Mj the j th homogeneous component of M . The elements x 2 Mj are said to behomogeneous (of degree j ).


Note that A0 is a commutative ring with 1 2 A0, that all summands Mj are A0-modules, and that M D L

j2ZMj is a direct sum decomposition of M as an A0-module.

Let A be a graded ring. The category of graded A-modules, denoted by GrA, hasas objects the graded A-modules. A morphism of graded A-modules f W M ! N isan A-module homomorphism satisfying f .Mj / � Nj for all j 2 Z. An A-modulehomomorphism which is a morphism in GrA will be called homogeneous.

Let M be a graded A-module and let N be a submodule of M . Say that N is agraded submodule if it is a graded module such that the inclusion map is a morphismin GrA. The graded submodules of A are called graded ideals. If d is an integer thetail M�d is the graded submodule of M having the same homogeneous components.M�d /j as M in degrees j d and zero for j < d . We also denote the ideal A�1by AC.

For n 2 Z, GrA comes equipped with a shift functor M 7! M.n/ where M.n/ isdefined by M.n/j D MnCj . Then GrA is a Grothendieck category with generatingfamily fA.n/gn2Z. The tensor product for the category of all A-modules induces atensor product on GrA: given two gradedA-modulesM;N and homogeneous elementsx 2 Mi ; y 2 Nj , set deg.x ˝ y/ WD i C j . We define the homomorphism A-moduleHomA.M;N / to be the graded A-module which is, in dimension n 2 Z, the groupHomA.M;N /n of graded A-module homomorphisms of degree n, i.e.,

HomA.M;N /n D GrA.M;N.n//:

We say that a graded A-module M is finitely generated if it is a quotient of a freegraded module of finite rank

LnsD1A.ds/where d1; : : : ; ds 2 Z. Say thatM is finitely

presented if there is an exact sequence

mMtD1

A.ct /!nMsD1

A.ds/!M ! 0:

The full subcategory of finitely presented graded modules will be denoted by grA. Notethat any graded A-module is a direct limit of finitely presented graded A-modules, andtherefore GrA is a locally finitely presented Grothendieck category.

LetE be any indecomposable injective gradedA-module (we remind the reader thatthe corresponding ungraded module,

LnEn, need not be injective in the category of

ungraded A-modules). Set P D P.E/ to be the sum of the annihilator ideals annA.x/of non-zero homogeneous elements x 2 E. Observe that each ideal annA.x/ is ho-mogeneous. Since E is uniform the set of annihilator ideals of non-zero homogeneouselements of E is upwards closed so the only issue is whether the sum, P.E/, of themall is itself one of these annihilator ideals.

Given a prime homogeneous idealP , we use the notationEP to denote the injectivehull, E.A=P /, of A=P . Notice that EP is indecomposable. We also denote the set ofisomorphism classes of indecomposable injective graded A-modules by InjA.


Lemma 5.1. IfE 2 InjA thenP.E/ is a homogeneous prime ideal. If the moduleE hasthe form EP .n/ for some prime homogeneous ideal P and integer n, then P D P.E/.Proof. The proof is similar to that of [14, 9.2].

It follows from the preceding lemma that the map

P � A 7! EP 2 InjA

from the set of homogeneous prime ideals to InjA is injective.A tensor torsion class in GrA is a torsion class S � GrA such that for any X 2 S

and any Y 2 GrA the tensor product X ˝ Y is in S .

Lemma 5.2. Let A be a graded ring. Then a torsion class S is a tensor torsion classof GrA if and only if it is closed under shifts of objects, i.e. X 2 S implies X.n/ 2 S

for any n 2 Z.

Proof. Suppose that S is a tensor torsion class of GrA. Then it is closed under shiftsof objects, because X.n/ Š X ˝ A.n/.

Assume the converse. Let X 2 S and Y 2 GrA. Then there is a surjectionLi2I A.i/

f� Y . It follows that 1X ˝ f W L

i2I X.i/ ! X ˝ Y is a surjection.Since each X.i/ belongs to S then so does X ˝ Y .

Lemma 5.3. The map

S 7! F.S/ D fa � A j A=a 2 Sgestablishes a bijection between the tensor torsion classes in GrA and the sets F ofhomogeneous ideals satisfying the following axioms:

T1: A 2 F;

T 2: if a 2 F and if a is a homogeneous element of A then it holds that .a W a/ Dfx 2 A j xa 2 ag 2 F;

T 3: if a and b are homogeneous ideals of A such that a 2 F and .b W a/ 2 F forevery homogeneous element a 2 a then b 2 F.

We shall refer to such filters as t -filters. Moreover, S is of finite type if and only if F.S/has a basis of finitely generated ideals, that is every ideal in F.S/ contains a finitelygenerated ideal belonging to F.S/. In this case F.S/ will be referred to as a t -filterof finite type.

Proof. It is enough to observe that there is a bijection between the Gabriel filters onthe family fA.n/gn2Z of generators closed under the shift functor (i.e., if a belongs tothe Gabriel filter then so does a.n/ for all n 2 Z) and the t -filters.

Proposition 5.4. The following statements are true:

1. Let F be a t -filter. If I; J belong to F, then IJ 2 F.


2. Assume that B is a set of homogeneous finitely generated ideals. The set B0 offinite products of ideals belonging to B is a basis for a t -filter of finite type.

Proof. 1. For any homogeneous element a 2 I we have .IJ W a/ � J , so IJ 2 F byT 3 and the fact that every homogeneous ideal containing an ideal from F must belongto F.

2. We follow [17, VI.6.10]. We must check that the set F of homogeneous idealscontaining ideals in B0 is a t -filter of finite type. T1 is plainly satisfied. Let a be ahomogeneous element inA and I 2 F. There is an ideal I 0 2 B0 contained in I . Then.I W a/ � I 0 and therefore .I W a/ 2 F, hence T 2 is satisfied as well.

Next we verify that F satisfies T 3. Suppose that I is a homogeneous ideal and thereexists J 2 F such that .I W a/ 2 F for every homogeneous element a 2 J . We mayassume that J 2 B0. Let a1; : : : ; an be generators of J . Then .I W ai / 2 F, i � n,and .I W ai / � Ji for some Ji 2 B0. It follows that aiJi � I for each i , and henceJJ1 : : : Jn � J.J1 \ � � � \ Jn/ � I , so I 2 F.

6 Torsion modules and the category QGr A

Let A be a commutative graded ring. In this section we introduce the category QGrA,which is analogous to the category of quasi-coherent sheaves on a projective variety.The non-commutative analog of the category QGrA plays a prominent role in “non-commutative projective geometry" (see, e.g., [1], [16], [19]).

Recall that the projective scheme ProjA is a topological space whose points are thehomogeneous prime ideals not containing AC. The topology of ProjA is defined bytaking the closed sets to be the sets of the form V.I / D fP 2 ProjA j P � I g for I ahomogeneous ideal of A. We set D.I/ WD ProjA n V.I /.

In the remainder of this section the homogeneous ideal AC � A is assumed tobe finitely generated. This is equivalent to assuming that A is a finitely generatedA0-algebra. The space ProjA is spectral and the quasi-compact open sets are thoseof the form D.I/ with I finitely generated (see, e.g., [9, 5.1]). Let TorsA denote thetensor torsion class of finite type corresponding to the family of homogeneous finitelygenerated ideals fAnCgn�1 (see Proposition 5.4). We refer to the objects of TorsA astorsion graded modules.

Let QGrA D GrA=TorsA. Let Q denote the quotient functor GrA ! QGrA.We shall identify QGrA with the full subcategory of Tors-closed modules. The shiftfunctor M 7!M.n/ defines a shift functor on QGrA for which we shall use the samenotation. Observe that Q commutes with the shift functor. Finally we shall writeO D Q.A/. Note that QGrA is a locally finitely generated Grothendieck categorywith the family, fQ.M/gM2grA, of finitely generated generators (see [7, 5.8]).

The tensor product in GrA induces a tensor product in QGrA, denoted by �. Moreprecisely, one sets

X � Y WD Q.X ˝ Y /for any X; Y 2 QGrA.


Lemma 6.1. Given X; Y 2 GrA there is a natural isomorphism in QGrA: Q.X/�Q.Y / Š Q.X ˝ Y /. Moreover, the functor �� Y W QGrA ! QGrA is right exactand preserves direct limits.

Proof. See [9, 4.2].

As a consequence of this lemma we get an isomorphism X.d/ Š O.d/ � X forany X 2 QGrA and d 2 Z.

The notion of a tensor torsion class of QGrA (with respect to the tensor product�) is defined analogously to that in GrA. The proof of the next lemma is like that ofLemma 5.2 (also use Lemma 6.1).

Lemma 6.2. A torsion class S is a tensor torsion class of QGrA if and only if it isclosed under shifts of objects, i.e. X 2 S implies X.n/ 2 S for any n 2 Z.

Given a prime ideal P 2 ProjA and a graded module M , denote by MP thehomogeneous localization ofM at P . If f is a homogeneous element of A, byMf wedenote the localization of M at the multiplicative set Sf D ff ngn�0.

Lemma 6.3. If T is a torsion module then TP D 0 and Tf D 0 for anyP 2 ProjA andf 2 AC. As a consequence, MP Š Q.M/P and Mf Š Q.M/f for any M 2 GrA.

Proof. See [9, 5.5].

Denote by Ltor.GrA;TorsA/ (respectively Ltor.QGrA/) the lattice of the tensortorsion classes of finite type in GrAwith torsion classes containing TorsA (respectivelythe tensor torsion classes of finite type in QGrA) ordered by inclusion. The map

` W Ltor.GrA;TorsA/! Ltor.QGrA/; S 7! S=TorsA

is a lattice isomorphism, where S=TorsA D fQ.M/ j M 2 Sg (see, e.g., [7, 1.7]).We shall consider the map ` as an identification.

Theorem 6.4 (Classification). Let A be a commutative graded ring which is finitelygenerated as an A0-algebra. Then the maps

V 7! S D fM 2 QGrA j suppA.M/ � V g and S 7! V D[M2S

suppA.M/

induce bijections between

1. the set of all open subsets V � Proj�A,

2. the set of all tensor torsion classes of finite type in QGrA.

Proof. By Lemma 5.3 it is enough to show that the maps

V 7! FV D fI 2 A j V.I / � V g and F 7! VF D[I2F

V.I /


induce bijections between the set of all open subsets V � Proj�A and the set of allt -filters of finite type containing fAnCgn�1.

Let F be such a t -filter. Then the set ƒF of finitely generated graded ideals Ibelonging to F is a basis for F. Clearly VF D

SI2ƒF

V.I /, so VF is open in Proj�A.Now let V be an open subset of Proj�A. Let ƒ be the set of finitely generated

homogeneous ideals I such that V.I / � V . Then V DSI2ƒ V.I / and I1 � � � In 2 ƒ

for any I1; : : : ; In 2 ƒ. We denote by F0V the set of homogeneous ideals I � A suchthat I � J for some J 2 ƒ. By Proposition 5.4(2) F0V is a t -filter of finite type.Clearly, F0V � FV . Suppose I 2 FV nF0V .

We can use Zorn’s lemma to find an ideal J � I which is maximal with respect toJ … F0V (we use the fact that F0V has a basis of finitely generated ideals). We claim thatJ is prime. Indeed, suppose a; b 2 A are two homogeneous elements not belonging toJ . ThenJCaA andJCbAmust be members of F0V , and also .JCaA/.JCbA/ 2 F0Vby Proposition 5.4(1). But .J C aA/.J C bA/ � J C abA, and therefore ab … J .We see that J 2 V.I / � V , and hence J 2 V.I 0/ for some I 0 2 ƒ. But this impliesJ 2 F0V , a contradiction. Thus F0V D FV . Clearly, V D VFV for every open subsetV � Proj�A. Let F be a t -filter of finite type and I 2 F. Then I � J for someJ 2 ƒF, and hence V.I / � V.J / � VF. It follows that F � FVF

. As above, there isno ideal belonging to FVF

nF. We have shown the desired bijection between the setsof all t -filters of finite type and all open subsets in Proj�A.

7 The prime spectrum of an ideal lattice

Inspired by recent work of Balmer [4], Buan, Krause, and Solberg [5] introduce thenotion of an ideal lattice and study its prime ideal spectrum. Applications arise fromabelian or triangulated tensor categories.

Definition (Buan, Krause, Solberg [5]). An ideal lattice is by definition a partiallyordered set L D .L;�/, together with an associative multiplication LL! L, suchthat the following holds.

(L1) The poset L is a complete lattice, that is,

supA D_a2A

a and inf A Dâ2A

a

exist in L for every subset A � L.

(L2) The latticeL is compactly generated, that is, every element inL is the supremumof a set of compact elements. (An element a 2 L is compact, if for all A � Lwith a � supA there exists some finite A0 � A with a � supA0.)

(L3) We have for all a; b; c 2 La.b _ c/ D ab _ ac and .a _ b/c D ac _ bc:


(L4) The element 1 D supL is compact, and 1a D a D a1 for all a 2 L.

(L5) The product of two compact elements is again compact.

A morphism � W L! L0 of ideal lattices is a map satisfying

�� _a2A

a�D

_a2A

�.a/ for A � L;

�.1/ D 1 and �.ab/ D �.a/�.b/ for a; b 2 L:

Let L be an ideal lattice. Following [5] we define the spectrum of prime elementsin L. An element p ¤ 1 in L is prime if ab � p implies a � p or b � p for alla; b 2 L. We denote by SpecL the set of prime elements in L and define for eacha 2 L

V.a/ D fp 2 SpecL j a � pg and D.a/ D fp 2 SpecL j a 6� pg:The subsets of SpecL of the formV.a/ are closed under forming arbitrary intersectionsand finite unions. More precisely,

V� _i2�

ai

�D

\i2�

V.ai / and V.ab/ D V.a/ [ V.b/:

Thus we obtain the Zariski topology on SpecL by declaring a subset of SpecL to beclosed if it is of the form V.a/ for some a 2 L. The set SpecL endowed with thistopology is called the prime spectrum of L. Note that the sets of the form D.a/ withcompact a 2 L form a basis of open sets. The prime spectrum SpecL of an ideallattice L is spectral [5, 2.5].

There is a close relation between spectral spaces and ideal lattices. Given a topo-logical space X , we denote by Lopen.X/ the lattice of open subsets of X and considerthe multiplication map

Lopen.X/ Lopen.X/! Lopen.X/; .U; V / 7! UV D U \ V:The lattice Lopen.X/ is complete.

The following result, which appears in [5], is part of the Stone Duality Theorem(see, for instance, [12]).

Proposition 7.1. Let X be a spectral space. Then Lopen.X/ is an ideal lattice. More-over, the map

X ! SpecLopen.X/; x 7! X n fxg;is a homeomorphism.

We deduce from the classification of torsion classes of finite type (Theorems 2.2and 6.4) the following.


Proposition 7.2. Let R (respectively A) be a commutative ring (respectively gradedcommutative ring which is finitely generated as an A0-algebra). Then Ltor.ModR/and Ltor.QGrA/ are ideal lattices.

Proof. The spaces SpecR and ProjA are spectral. Thus Spec�R and Proj�A arespectral, also Lopen.Spec�R/ and Lopen.Proj�A/ are ideal lattices by Proposition 7.1.By Theorems 2.2 and 6.4 we have isomorphisms Lopen.Spec�R/ Š Ltor.ModR/ andLopen.Proj�A/ Š Ltor.QGrA/. Therefore Ltor.ModR/ and Ltor.QGrA/ are ideallattices.

Corollary 7.3. Let R (respectively A) be a commutative ring (respectively the gradedcommutative ring which is finitely generated as an A0-algebra). The points ofSpecLtor.ModR/ (respectively SpecLtor.QGrA/) are the \-irreducible torsionclasses of finite type in ModR (respectively tensor torsion classes of finite type inQGrA) and the maps

f W Spec�R! SpecLtor.ModR/; P 7! SP D fM 2 ModR jMP D 0g;f W Proj�A! SpecLtor.QGrA/; P 7! SP D fM 2 QGrA jMP D 0g

are homeomorphisms of spaces.

Proof. This is a consequence of Theorems 2.2, 6.4 and Propositions 7.1, 7.2.

8 Reconstructing affine and projective schemes

Let R (respectively A) be a commutative ring (respectively graded commutative ringwhich is finitely generated as an A0-algebra). We shall write Spec.ModR/ WDSpec�Ltor.ModR/ (resp. Spec.QGrA/ WD Spec�Ltor.QGrA/) and supp.M/ WDfP 2 Spec.ModR/ j M 62 P g (resp. supp.M/ WD fP 2 Spec.QGrA/ j M 62 P g)for M 2 ModR (resp. M 2 QGrA). It follows from Corollary 7.3 that

suppR.M/ D f �1.supp.M// (resp. suppA.M/ D f �1.supp.M//):

Following [4], [5], we define a structure sheaf on Spec.ModR/ (Spec.QGrA/) asfollows. For an open subset U � Spec.ModR/ (U � Spec.QGrA/), let

SU D fM 2 ModR .QGrA/ j supp.M/ \ U D ;gand observe that SU D fM jMP D 0 for all P 2 f �1.U /g is a (tensor) torsion class.We obtain a presheaf of rings on Spec.ModR/ (Spec.QGrA/) by

U 7! EndModR=SU .R/ .EndQGrA=SU .O//:

If V � U are open subsets, then the restriction map

EndModR=SU .R/! EndModR=SV .R/ .EndQGrA=SU .O/! EndQGrA=SV .O//


is induced by the quotient functor

ModR=SU ! ModR=SV .QGrA=SU ! QGrA=SV /:

The sheafification is called the structure sheaf of ModR (QGrA) and is denoted byOModR (OQGrA). This is a sheaf of commutative rings by [13, XI.2.4]. Next letP 2 Spec.ModR/ and P WD f �1.P /. We have

OModR;P D lim�!P2U

EndModR=SU .R/ D lim�!f …P

EndModR=SD.f /.R/ Š lim�!f …P

Rf D OR;P :

Similarly, for P 2 Spec.QGrA/ and P WD f �1.P / we have

OQGrA;P Š OProjA;P :

The next theorem says that the abelian category ModR (QGrA) contains all thenecessary information to reconstruct the affine (projective) scheme .SpecR;OR/ (re-spectively .ProjA;OProjA/).

Theorem 8.1 (Reconstruction). Let R (respectively A) be a ring (respectively gradedring which is finitely generated as an A0-algebra). The maps of Corollary 7.3 induceisomorphisms of ringed spaces

f W .SpecR;OR/��! .Spec.ModR/;OModR/

and

f W .ProjA;OProjA/��! .Spec.QGrA/;OQGrA/:

Proof. The proof is similar to that of [5, 9.4]. Fix an open subset U � Spec.ModR/and consider the composition of the functors

F W ModR�.�/��! Qcoh SpecR

.�/jf�1.U/��! Qcoh f �1.U /:

Here, for any R-module M , we denote by �M its associated sheaf. By definition,the stalk of �M at a prime P equals the localized module MP . We claim that Fannihilates SU . In fact,M 2 SU implies f �1.supp.M//\f �1.U / D ; and thereforesuppR.M/\f �1.U / D ;. ThusMP D 0 for allP 2 f �1.U / and thereforeF.M/ D0. It follows thatF factors through ModR=SU and induces a map EndModR=SU .R/!OR.f

�1.U // which extends to a map OModR.U / ! OR.f�1.U //. This yields the

morphism of sheaves f ] W OModR ! f�OR.By the above f ] induces an isomorphism f

]P W OModR;f .P / ! OR;P at each point

P 2 SpecR. We conclude that f ]P is an isomorphism. It follows that f is an isomor-phism of ringed spaces if the map f W SpecR! Spec.ModR/ is a homeomorphism.This last condition is a consequence of Propositions 7.1 and 7.2. The same argumentsapply to show that

f W .ProjA;OProjA/��! .Spec.QGrA/;OQGrA/

is an isomorphism of ringed spaces.


9 Appendix

A subcategory S of a Grothendieck category C is said to be Serre if for any short exactsequence

0! X 0 ! X ! X 00 ! 0

X 0; X 00 2 S if and only if X 2 S . A Serre subcategory S of C is said to be atorsion class if S is closed under taking coproducts. An object C of C is said to betorsionfree if .S ; C / D 0. The pair consisting of a torsion class and the correspondingclass of torsionfree objects is referred to as a torsion theory. Given a torsion class S

in C the quotient category C=S is the full subcategory with objects those torsionfreeobjects C 2 C satisfying Ext1.T; C / D 0 for every T 2 S . The inclusion functori W S ! C admits the right adjoint t W C ! S which takes every object X 2 C to themaximal subobject t .X/ ofX belonging to S . The functor t we call the torsion functor.Moreover, the inclusion functor i W C=S ! C has a left adjoint, the localization functor.�/S W C ! C=S , which is also exact. Then,

HomC .X; Y / Š HomC=S .XS ; Y /

for allX 2 C and Y 2 C=S . A torsion class S is of finite type if the functor i W C=S !C preserves directed sums. If C is a locally coherent Grothendieck category then S isof finite type if and only if i W C=S ! C preserves direct limits (see, e.g., [7]).

Let C be a Grothendieck category having a family of finitely generated projectivegenerators A D fPigi2I . Let F DS

i2I Fi be a family of subobjects, where each Fi

is a family of subobjects ofPi . We refer to F as a Gabriel filter if the following axiomsare satisfied:

T1: Pi 2 Fi for every i 2 I ;

T 2: if a 2 Fi and � W Pj ! Pi then fa W �g D ��1.a/ 2 Fj ;

T 3: if a and b are subobjects of Pi such that a 2 Fi and fb W �g 2 Fj for any� W Pj ! Pi with Im� � a then b 2 Fi .

In particular each Fi is a filter of subobjects of Pi . A Gabriel filter is of finite type ifeach of these filters has a cofinal set of finitely generated objects (that is, if for each iand each a 2 Fi there is a finitely generated b 2 Fi with a � b).

Note that if A D fAg is a ring and a is a right ideal of A, then for every endomor-phism � W A! A

��1.a/ D fa W �.1/g D fa 2 A j �.1/a 2 ag:On the other hand, if x 2 A, then fa W xg D ��1.a/, where � 2 EndA is such that�.1/ D x.

It is well-known (see, e.g., [7]) that the map

S 7�! F.S/ D fa � Pi j i 2 I; Pi=a 2 Sgestablishes a bijection between the Gabriel filters (respectively Gabriel filters of finitetype) and the torsion classes on C (respectively torsion classes of finite type on C ).


References

[1] M. Artin, J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109(2) (1994),228–287.

[2] M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra, Addison-WesleyPublishing Co., Reading, Mass, 1969.

[3] P. Balmer, Presheaves of triangulated categories and reconstruction of schemes, Math. Ann.324(3) (2002), 557–580.

[4] P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew.Math. 588 (2005), 149–168.

[5] A. B. Buan, H. Krause, Ø. Solberg, Support varieties – an ideal approach, Homology,Homotopy Appl. 9 (2007), 45–74.

[6] P. Gabriel, Des catégories abeliénnes, Bull. Soc. Math. France 90 (1962), 323–448.

[7] G. Garkusha, Grothendieck categories, Algebra i Analiz 13(2) (2001), 1–68; English transl.St. Petersburg Math. J. 13 (2002), 149–200.

[8] G. Garkusha, M. Prest, Classifying Serre subcategories of finitely presented modules, Proc.Amer. Math. Soc. 136 (2008), 761–770.

[9] G. Garkusha, M. Prest, Reconstructing projective schemes from Serre subcategories, J. Al-gebra 319 (2008), 1132–1153.

[10] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142(1969), 43–60.

[11] M. Hovey, Classifying subcategories of modules, Trans. Amer. Math. Soc. 353 (2001),3181—3191.

[12] P. T. Johnstone, Stone Spaces, Cambridge Stud. in Adv. Math. 3, Cambridge UniversityPress, Cambridge 1982.

[13] Ch. Kassel, Quantum groups, Grad. Texts in Math. 155, Springer-Verlag, New York 1995.

[14] M. Prest, The Zariski spectrum of the category of finitely presented modules, preprintavailable at maths.man.ac.uk/�mprest.

[15] A. L. Rosenberg, The spectrum of abelian categories and reconstruction of schemes, inRings, Hopf algebras, and Brauer groups, Lect. Notes PureAppl. Math. 197, Marcel Dekker,New York 1998, 257–274.

[16] J. T. Stafford, Noncommutative projective geometry, in Proceedings of the InternationalCongress of Mathematicians (Beijing, 2002), Vol. II, Higher Ed. Press, Beijing 2002,93–103.

[17] B. Stenström, Rings of quotients, Grundlehren Math. Wiss. 217, Springer-Verlag, Berlin,Heidelberg, New York 1975.

[18] R. W. Thomason, The classification of triangulated subcategories, Compos. Math. 105(1997), 1–27.

[19] A. B. Verevkin, On a noncommutative analogue of the category of coherent sheaves on aprojective scheme, in Algebra and analysis (Tomsk, 1989), Amer. Math. Soc. Transl. (2)151, Amer. Math. Soc., Providence, RI, 1992.

[20] M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), 149–213.

Parshin’s conjecture revisited

Thomas Geisser�

1 Introduction

Parshin’s conjecture states that Ki .X/Q D 0 for i > 0 and X smooth and projectiveover a finite field Fq . The purpose of this paper is to break up Parshin’s conjectureinto several independent statements, in the hope that each of them is easier to attackindividually. If CHn.X; i/ is Bloch’s higher Chow group of cycles of relative dimen-sion n, then in view ofKi .X/Q ŠL

n CHn.X; i/Q, Parshin’s conjecture is equivalentto ConjectureP.n/ for all n, stating that CHn.X; i/Q D 0 for i > 0, and all smooth andprojective X . We show, assuming resolution of singularities, that Conjecture P.n/ isequivalent to the conjunction of three conjecturesA.n/,B.n/ andC.n/, and give severalequivalent versions of these conjectures. This is most conveniently formulated in termsof weight homology. We define HW� .X;Q.n// to be the homology of the complexCHn.W.X//Q, whereW.X/ is the weight complex defined by Gillet–Soulé [8] shiftedby 2n. Then, in a nutshell, Conjecture A.n/ states that for all schemes X over Fq ,the niveau spectral sequence of CHn.X;�/Q degenerates to one line, Conjecture C.n/states that for all schemes X over Fq the niveau spectral sequence of HW� .X;Q.n//degenerates to one line, and Conjecture B.n/ states that, for all X over Fq , the twolines are isomorphic. The conjunction of A.n/, B.n/, and C.n/ clearly implies P.n/,because HW

i .X;Q.n// D 0 for i 6D 2n and X smooth and projective over Fq , and weshow the converse. Note that in the above formulation, Parshin’s conjecture impliesstatements on higher Chow groups not only for smooth and projective schemes, butgives a way to calculate CHn.X; i/Q for arbitrary X .

A more concrete version ofA.n/ is that, for every smooth and projective schemeXof dimension d over Fq , CHn.X; i/Q D 0 for i > d � n. A reformulation of B.n/ isthat for every smooth and projective schemeX of dimension d > nC 1, the followingsequence is exact

0! KMd�n.k.X//Q !Mx2X.1/

KMd�n�1.k.x//Q !Mx2X.2/

KMd�n�2.k.x//Q

and that this sequence is exact at KM1 .k.X//Q if d D dimX D nC 1.In the second half of the paper, we focus on the case n D 0, because of its applica-

tions in [5]. A different version of weight homology has been studied by Jannsen [11],and he proved that Conjecture C.0/ holds under resolution of singularities. We use thisto give two more versions of Conjecture P.0/. The first is that there is an isomorphism

�Supported in part by NSF grant No.0556263

414 T. Geisser

from higher Chow groups with Ql -coefficients to l-adic cohomology, for all X and i ,

CH0.X; i/Ql ˚ CH0.X; i C 1/Ql ! Hi .Xet; yQl/:

ConjectureP.0/ can also be recovered from, and implies a structure theorem for higherChow groups of smooth affine schemes: For all smooth and affine schemes U ofdimension d over Fq , the groups CH0.U; i/ are torsion for i 6D d , and the canonicalmap CH0.U; d/Ql ! Hd . xUet; yQl/

Gal.Fq/ is an isomorphism.Finally, we reproduce an argument of Levine showing that if F is the absolute

Frobenius, the push-forward F� acts like qn on CHn.X; i/, and the pull-back F � actson motivic cohomology H i

M .X;Z.n// like qn for all n. As a corollary, ConjectureP.0/ follows from finite dimensionality of smooth and projective schemes over finitefields in the sense of Kimura [13].

Acknowledgements. This paper was inspired by the work of, and discussions with,U. Jannsen and S. Saito. We are indebted to the referee, whose careful reading helpedto improve the exposition.

2 Parshin’s conjecture

We fix a perfect field k of characteristic p, and consider the category of separatedschemes of finite type over k. We recall some facts on Bloch’s higher Chow groups[1], see [3] for a survey. Let zn.X; i/ be the free abelian group generated by cycles ofdimension n C i on X �k �i which meet all faces properly, and let zn.X;�/ be thecomplex of abelian groups obtained by taking the alternating sum of intersection withface maps as differential. We define CHn.X; i/ as the i th homology of this complexand motivic Borel–Moore homology to be

H ci .X;Z.n// D CHn.X; i � 2n/:

For a proper map f W X ! Y we have a push-forward zn.X;�/! zn.Y;�/, for a flat,quasi-finite map f W X ! Y we have a pull-back zn.X;�/ ! zn.Y;�/, and a closedembedding i W Z ! X with open complement j W U ! X induces a localizationsequence

� � � ! H ci .Z;Z.n//

i��! H ci .X;Z.n//

j�

�! H ci .U;Z.n//! � � � :

IfX is smooth of pure dimension d , thenH ci .X;Z.n// Š H 2d�i .X;Z.d�n//, where

the right-hand side is Voevodsky’s motivic cohomology [15]. For a finitely generatedfield F over k, we define H c

i .F;Z.n// D colimU Hci .U;Z.n//, where the colimit

runs through U of finite type over k with field of functions F . For the reader who ismore familiar with motivic cohomology, we mention that Voevodsky’s theorem impliesthat for a field F of transcendence degree d over k, we have

H ci .F;Z.n// Š H 2d�i .F;Z.d � n// Š

(0; i < d C n;KMd�n.F /; i D d C n:

Parshin’s conjecture revisited 415

The latter isomorphism is due to Nesterenko–Suslin and Totaro. It follows formallyfrom localization that there are spectral sequences

E1s;t .X/ DMx2X.s/

H csCt .k.x/;Z.n//) H c

sCt .X;Z.n//: (1)

Here X.s/ denotes points of X of dimension s. Since H ci .F;Z.n// D 0 for i <

n C trdegF , the spectral sequence is concentrated in the area 0 � s � dimX andt � n. If we let

zH ci .X;Z.n// D E2iCn;n.X/

be the i th homology of the complex

0 Mx2X.n/

H c2n.k.x/;Z.n// � � �

Mx2X.s/

H csCn.k.x/;Z.n// � � � ; (2)

withLx2X.s/ H

csCn.k.x/;Z.n// in degree s C n, then we obtain a canonical and

functorial map˛ W H c

i .X;Z.n//! zH ci .X;Z.n//:

Note that the groups in (2) are Milnor-K-groups.Parshin’s conjecture states that for all smooth and projective X over Fq , the groups

Ki .X/Q are torsion for i > 0. If Tate’s conjecture holds and rational equivalenceand homological equivalence agree up to torsion for all X , then Parshin’s conjectureholds by [2]. Since Ki .X/Q DL

n CHn.X; i/Q, it follows that Parshin’s conjectureis equivalent to the following conjecture for all n.

Conjecture P.n/. For all smooth and projective schemes X over the finite field Fq ,the groups H c

i .X;Q.n// vanish for i 6D 2n.

We will refer to the following equivalent statements as Conjecture A.n/:

Proposition 2.1. For a fixed integer n, the following statements are equivalent:

a) For all schemes X=Fq and all i , ˛ induces an isomorphism H ci .X;Q.n// ŠzH c

i .X;Q.n//.

b) For all finitely generated fields k=Fq with d WD trdeg k=Fq , and all i 6D d C n,we have H c

i .k;Q.n// D 0.

c) For all smooth and projective X over Fq and all i > dimX C n, we haveH ci .X;Q.n// D 0.

d) For all smooth and affine schemes U over Fq and all i > dimU C n, we haveH ci .U;Q.n// D 0.

Proof. a)) c), d): The complex (2) is concentrated in degrees Œ2n; d C n�.c)) b): This is proved by induction on the transcendence degree. By de Jong’s

theorem, we find a smooth and proper model X of a finite extension of k. Look-ing at the niveau spectral sequence (1), we see that the induction hypothesis impliesH ci .X;Q.n// D 0 for i > d C n (see [2] for details).

416 T. Geisser

d) ) b): follows by writing k as a colimit of smooth affine scheme schemes ofdimension d .

b)) a): The niveau spectral sequence collapses to the complex (2).

Using the Gersten resolution, the statement in the proposition implies that on asmooth X of dimension d , the motivic complex Q.d � n/ is concentrated in degreed � n, and if Cn WD Hd�n.Q.d � n// D CHn.�; d � n/Q, then CHn.X; i/ DHd�n�i .X;Cn/.

Statement 2.1 d) is part of the following affine analog of P.n/:

Conjecture L.n/. For all smooth and affine schemes U of dimension d over the finitefield Fq , the group H c

i .U;Q.n// vanishes unless d � i � d C n.

Since H ci .U;Q.n// Š H 2d�i .U;Q.d � n//, Conjecture L.n/ can be thought of

as an analog of the affine Lefschetz theorem.

3 Weight homology

This section is inspired by Jannsen [11]. Throughout this section we assume resolutionof singularities over the field k. Let C be the category with objects smooth projectivevarieties over a field k of characteristic 0, and HomC .X; Y / DL

i CHdimYi .X � Yi /,where Yi runs through the connected components of Y . Let H be the homotopycategory of bounded complexes over C . Gillet and Soulé define in [8], for everyseparated scheme of finite type, a weight complexW.X/ 2 H satisfying the followingproperties [8, Theorem 2] (our notation differs from loc. cit. in variance):

a) W.X/ is represented by a bounded complex

M.X0/ M.X1/ � � � M.Xk/

with dimXi � dimX � i , where M.Xi / placed in degree i .

b) W.�/ is covariant functorial for proper maps.

c) W.�/ is contravariant functorial for open embeddings.

d) If T ! X is a closed embedding with open complement U , then there is adistinguished triangle

W.T /i��! W.X/

j�

�! W.U /:

e) If D is a divisor with normal crossings in a scheme X , with irreducible compo-nents Yi , and if Y .r/ D`

#IDr \i2IYi , then W.X �D/ is represented by

M.X/ M.Y .1// � � � M.Y .dimX//:

The argument in loc. cit. only uses that resolution of singularities exists over k, andwe assume from now on that k is such a field.


Given an additive covariant functor F from C to an abelian category, we defineweight (Borel–Moore) homology HW

i .X; F / as the i th homology of the complexF.W.X//. Weight homology has the functorialities inherited from b) and c), andsatisfies a localization sequence deduced from d). If K is a finitely generated fieldover k, then we define HW

i .K; F / to be colimHWi .U; F /, where the (filtered) limit

runs through integral varieties havingK as their function field. Similarly, a contravariantfunctorG fromC to an abelian category gives rise to weight cohomology (with compactsupport) H i

W .X;G/.As a special case, we define the weight homology group HW

i .X;Z.n// as thei � 2nth homology of the homological complex of abelian groups CHn.W.X//.

Lemma 3.1. We have HWi .X;Z.n// D 0 for i > dimX C n. In particular we have

HWi .K;Z.n// D 0 for every finitely generated fieldK=k and every i > trdegk KCn.

Proof. This follows from the first property of weight complexes together withCHn.T / D 0 for n > dim T .

It follows from Lemma 3.1 that the niveau spectral sequence

E1s;t .X/ DMx2X.s/

HWsCt .k.x/;Z.n//) HW

sCt .X;Z.n// (3)

is concentrated on and below the line t D n. Let

zHWi .X;Z.n// D E2iCn;n.X/

be the i th homology of the complex

0 Mx2X.n/

HW2n.k.x/;Z.n// � � �

Mx2X.s/

HWsCn.k.x/;Z.n// � � � ; (4)

whereLx2X.s/ H

WsCn.k.x/;Z.n// is placed in degree sCn. Then we obtain a canonical

and natural map� W zHW

i .X;Z.n//! HWi .X;Z.n//:

Consider the canonical map of covariant functors � 0 W zn.�;�/! CHn.�/ on thecategory of smooth projective schemes over k, sending the cycle complex to its highestcohomology. Then by [11, Theorem 5.13, Remark 5.15], the set of associated homologyfunctors extends to a homology theory on the category of all varieties over k. Theargument of [11, Proposition 5.16] show that the extension of the associated homologyfunctors for zn.�;�/ are higher Chow groups CHn.�; i/. The extension CHn.�/ areby definition the functors HW

i .�;Z.n//. We obtain a functorial map

� W H ci .X;Z.n//! HW

i .X;Z.n//:

Lemma 3.2. For i D 2n, and for all schemes X , the map � is an isomorphismH c2n.X;Z.n// Š HW

2n.X;Z.n//. In particular,HW2d.K;Z.d// Š Z for all fieldsK of

transcendence degree d over k.

418 T. Geisser

Proof. The statement is clear forX smooth and projective. We proceed by induction onthe dimension of X . Using the localization sequence for both theories, we can assumethatX is proper. Let f W X 0 ! X be a resolution of singularities ofX ,Z be the closedsubscheme (of lower dimension) where f is not an isomorphism, and Z0 D Z �X X 0.Then we conclude by comparing localization sequences

H c2n.Z

0;Z.n// �� H c2n.Z;Z.n//˚H c

2n.X0;Z.n// �� H c

2n.X;Z.n// ��

��

0

HW2n.Z

0;Z.n// �� HW2n.Z;Z.n//˚HW

2n.X0;Z.n// �� HW

2n.X;Z.n//�� 0.

The map � for fields induces a map ˇ W zH ci .X;Q.n//! zHW

i .X;Q.n//, which fitsinto the (non-commutative) diagram

H ci .X;Z.n//

� ��

˛

��

HWi .X;Z.n//

zH ci .X;Z.n//

ˇ�� zHW

i .X;Z.n//.

�

��

We now return to the situation k D Fq , and compare weight homology to higherChow groups using their niveau spectral sequences. We saw that the niveau spectralsequence for higher Chow groups is concentrated above the line t D n, and that theniveau spectral sequence for weight homology is concentrated below the line t D n.Our aim is to show that Parshin’s conjecture is equivalent to both being rationallyconcentrated on this line, and that the resulting complexes are isomorphic.

The following statements will be referred to as Conjecture B.n/:


a) The map ˇ induces an isomorphism zHi .X;Q.n// Š zHWi .X;Q.n// for all

schemes X and all i .

b) The map � induces an isomorphism H cdCn.k;Q.n// Š HW

dCn.k;Q.n// for allfinitely generated fields k=Fq , where d D trdeg k=Fq .

c) For every smooth and projectiveX over Fq the following holds: if d D dimX >

nC 1, then zH cdCn.X;Q.n// D zH c

dCn�1.X;Q.n// D 0, and if dimX D nC 1,

then zH c2nC1.X;Q.n// D 0.

Note that assuming the conjecture A.n/, c) is equivalent to H cdCn.X;Q.n// D

H cdCn�1.X;Q.n// D 0 and H c

2nC1.X;Q.n// D 0 for all smooth and projective X ofdimension d > nC 1 and d D nC 1, respectively, hence are part of Conjecture P.n/.


Proof. b)) a) is trivial, and a)) b) follows by a colimit argument because

colimU�X

zH ci .X;Q.n// Š

(H cdCn.k.X/;Q.n//; i D d C nI

0; otherwise.

a)) c) follows because for X smooth and projective of dimension d , the coho-mology of the complex (2) tensored with Q equals zH c

i .X;Q.n// D zHWi .X;Q.n// for

i D d C n and i D d C n � 1 (or i D 2nC 1 in case d D nC 1). An inspection ofthe niveau spectral sequence (3) shows that this is a subgroup of HW

i .X;Q.n// D 0.c) ) b): For n > d , both sides vanish, whereas for d D n, both sides are

canonically isomorphic to Q. For n < d , we proceed by induction on d . Choose asmooth and projective model X for k and compare the exact sequences (2) and (4)

ALx2X.d�1/

H cdCn�1.k.x/;Q.n//�� H c

dCn.k;Q.n//��

��

BLx2X.d�1/

HWdCn�1.k.x/;Q.n//�� HW

dCn.k;Q.n//.��

The terms on the left areA D H c2n.X;Q.n// Š B D HW

2n.X;Q.n// if d D nC1, andA D L

x2X.d�2/H cdCn�2.k.x/;Q.n// Š B D L

x2X.d�2/HWdCn�2.k.x/;Q.n//, if

d > nC 1. The upper sequence is exact by hypothesis, and an inspection of (3) showsthat the lower sequence is exact because HW

i .X;Q.n// D 0 for i > 2n.

We refer to the following statements as Conjecture C.n/:


a) For all schemes X over Fq and for all i , the map � induces an isomorphismzHWi .X;Q.n/ Š HW

i .X;Q.n//.

b) For all finitely generated fields k=Fq and for all i 6D trdeg k=Fq C n, we haveHWi .k;Q.n// D 0.

c) For all smooth and projective X , the map � induces an isomorphism

zHWi .X;Q.n// D

(0; i > 0ICHn.X/Q; i D 2n:

Proof. b)) a) is trivial and a)) b) follows by a colimit argument.a)) c) is trivial for i > 2n, and Lemma 3.2 for i D 2n.c) ) b): This is proved like Proposition 2.1, by induction on the transcendence

degree of k. Let X be a smooth and projective model of k. The induction hypothesisimplies that the niveau spectral sequence (3) collapses to the horizontal line t D n

and the vertical line s D d . Since it converges to HWi .X;Q.n//, which is zero for

i > 2n, we obtain isomorphisms HWdCn�iC1.k.X/;Q.n//

di�! zHWdCn�i .X;Q.n// for

420 T. Geisser

1 < i < d � n, and an exact sequence

HW2nC1.k.X/;Q.n//

dd�n,! zHW

2n.X;Q.n//! HW2n.X;Q.n// � HW

2n.k.X/;Q.n//:

The claim follows because zHWi .X;Q.n// D HW

i .X;Q.n// D 0 for i > 2n, and be-

cause the mapsH c2n.X;Q.n//

��! HW2n.X;Q.n//

� � zHW2n.X;Q.n// are isomorphisms

by Lemma 3.2 and hypothesis.

Theorem 3.5. The conjunction of Conjectures A.n/, B.n/ and C.n/ is equivalent toP.n/.

Proof. Given Conjectures A.n/, B.n/, and C.n/, we get

H ci .X;Q.n// D zH c

i .X;Q.n// Š zHWi .X;Q.n// Š HW

i .X;Q.n//

for all X , and the latter vanishes for smooth and projective X and i > 0, henceConjecture P.n/ follows. Conversely, Conjecture P.n/ implies Proposition 2.1 c),then 3.3 c), and finally 3.4 c) by using 2.1 a) and 3.3 a).

Remark. Propositions 2.1, 3.4, 3.3 as well as Theorem 3.5 remain true if we restrictourselves to schemes of dimension at most N , and to fields of transcendence degree atmost N , for a fixed integer N .

Remark. Gillet announced that one can obtain a rational version of weight complexesby using de Jong’s theorem on alterations instead of resolution of singularities. Thesame argument should then give the generalization [11, Theorem 5.13]. In this case,all arguments of this section hold true rationally, except the proof of c)) b) in Propo-sitions 3.3 and 3.4, and the proof of P.n/) A.n/; B.n/; C.n/ in Theorem 3.5, whichrequire that every finitely generated field over Fq has a smooth and projective model.

4 The case n D 0

Since H ci .X;Q.0// D CH0.X; i/Q, we use higher Chow groups in this section.

Proposition 4.1. We have CH0.X; i/Q Š zH ci .X;Q.0// for i � 2, and the map

CH0.X; 3/Q ! zH c3 .X;Q.0// is surjective for all X . In particular, A.0/ holds in

dimensions at most 2.

Proof. Since H i .k.x/;Q.1// D 0 for i 6D 1 and H i .k.x/;Q.0// D 0 for i 6D 0, thisfollows from an inspection of the niveau spectral sequence.

Jannsen [11] defines a variant of weight homology with coefficientsA,HWi .X;A/,

as the homology of the complex Hom.CH0.W.X//; A/. Note that HWi .X;Q/ D

HWi .X;Q.0// because Hom.CH0.X/;Q/ Š CH0.X/Q for smooth and projectiveX ,

in a functorial way. Indeed, a map of connected, smooth and projective varieties inducesthe identity pull-back on CH0 and the identity push-forward on CH0.


Theorem 4.2 (Jannsen). Under resolution of singularities, HWa .k; A/ D 0 for a 6D

trdeg k=Fq , hence HWi .X;A/ is the homology of the complex

0 Mx2X.0/

HW0 .k.x/; A/ � � �

Mx2X.s/

HWs .k.x/; A/ � � � (5)

for all schemes X . In particular, Conjecture C.0/ holds.

This is proved in [11, Proposition 5.4, Theorem 5.10]. The proof only works forn D 0, because it uses the bijectivity of CH0.Y /! CH0.X/ for a map of connectedsmooth and projective schemesX ! Y . The second statement follows using the niveauspectral sequence, which exists because HW

i .X;A/ satisfies the localization propertyby property (4) of weight complexes.

Let Zc.0/ be the complex of etale sheaves z0.�;�/. For any prime l , considerl-adic cohomology

Hi .Xet; yQl/ WD Q˝Z limHi .Xet;Zc=lr.0//:

In [4], we showed that for every positive integerm, and every scheme f W X ! k overa perfect field, there is a quasi-isomorphism Zc=m.0/ Š Rf ŠZ=m. In particular, theabove definition agrees with the usual definition of l-adic homology if l 6D p D char Fq .If xX D X �Fq

xFq and yG D Gal.xFq=Fq/, then there is a short exact sequence

0! HiC1. xXet; yQl/ OG ! Hi .Xet; yQl/! Hi . xXet; yQl/OG ! 0

and for U affine and smooth, Hi .Uet; yQl/ vanishes for i 6D d; d � 1 and l 6D p bythe affine Lefschetz theorem and a weight argument [12, Theorem 3a)]. The map fromZariski-hypercohomology of Zc=m.0/ to etale-hypercohomology of Zc=m.0/ inducesa functorial map

CH0.X; i/=m! CH0.X; i;Z=m/! Hi .Xet;Zc=m.0//;

hence in the limit a map

! W CH0.X; i/Ql ! Hi .Xet; yQl/:

Similarly, the map x! W CH0. xX; i/Ql ! Hi . xXet; yQl/ induces a map

� W CH0.X; i C 1/Ql� � .CH0. xX; i C 1/Ql / OG ! HiC1. xXet; yQl/ OG ! Hi .Xet; yQl/

(the left map is an isomorphism by a trace argument). The sum

'iX W CH0.X; i/Ql ˚ CH0.X; i C 1/Ql ! Hi .Xet; yQl/ (6)

is compatible with localization sequences, because all maps involved in the definitionare. The following proposition shows that Parshin’s conjecture can be recovered fromand implies a structure theorem for higher Chow groups of smooth affine schemes;compare to Jannsen [11, Conjecture 12.4b)].

422 T. Geisser

Proposition 4.3. The following statements are equivalent:

a) Conjecture P.0/.

b) For all schemes X , and all i , the map 'iX is an isomorphism.

c) For all smooth and affine schemes U of dimension d , the groups CH0.U; i/ aretorsion for i 6D d , and the composition ! W CH0.U; d/Ql ! Hd .Uet; yQl/ !Hd . xUet; yQl/

OG is an isomorphism.

Proof. a)) b): First consider the case that X is smooth and proper. Then ConjectureP(0) is equivalent to the vanishing of the left-hand side of (6) for i 6D 0;�1, whereas theright-hand side of (6) vanishes by the Weil-conjectures. On the other hand, '0X inducesan isomorphism CH0.X/˝Ql Š H 2d .Xet; yQl.d// and '�1X induces an isomorphismCH0.X/ ˝ Ql Š .CH0. xX/ ˝ Ql/ OG Š H 2d . xX; yQl.d// OG . Indeed, both sides areisomorphic to Ql if X is connected. Using localization and alterations, the statementfor smooth and proper X implies the statement for all X .

b)) a): The right-hand side of (6) is zero for i 6D 0;�1 by weight reasons forsmooth and projective X , hence so is the left side.

b)) c): This follows because Hi .Uet; yQl/ D 0 unless i D d; d � 1 for smoothand affine U .

c)) b): We first assume thatX is smooth and affine. By hypothesis and the affineLefschetz theorem, both sides of (6) vanish for i 6D d; d � 1, and are isomorphic fori D d . For i D d � 1, the vertical maps in the following diagram are isomorphismsby semi-simplicity,

CH0.X; d/Ql Š CH0. xX; d/ OGQl �� Hd . xX; yQl/OG

.CH0. xX; d/Ql / OG �� Hd . xX; yQl/ OG� �� Hd�1.X; yQl/.

Hence the lower map 'd�1X is an isomorphism because the upper map is. Using local-ization, the statement for smooth and affine X implies the statement for all X .

Proposition 4.4. Under resolution of singularities, the following are equivalent:

a) Conjecture P.0/.

b) For every smooth affine U of dimension d over Fq , we have CH0.U; i/Q ŠHWi .U;Q/ for all i , and these group vanish for i 6D d .

c) For every smooth affineU of dimension d over Fq , the groups CH0.U; i/Q vanishfor i > d , and CH0.U; d/Q Š HW

d.U;Q/.

Proof. a) ) b): It follows from the previous proposition that CH0.U; i/Q D 0 fori 6D d . On the other hand, P.0/ for all X implies that CH0.X; i/ Š HW

i .X;Q/ forall i and X .

c)) a): The statement implies Conjecture A.0/, and then Conjecture B.0/, ver-sion b), for all X . By Theorems 3.5 and 4.2, P.0/ follows.


Proposition 4.5. ConjectureP.0/ for all smooth and projectiveX implies the followingstatements:

a) (Affine Gersten) For every smooth affine U of dimension d , the following se-quence is exact:

CH0.U; d/Q ,!Mx2U .0/

Hd .k.x/;Q.d//!Mx2U .1/

Hd�1.k.x/;Q.d�1//!� � � :

b) Let X D Xd � Xd�1 � � � �X1 � X0 be a filtration such that Ui D Xi � Xi�1is smooth and affine of dimension i . Then CH0.X; i/Q is isomorphic to the i thhomology of the complex

0! CH0.Ud ; d /Q ! CH0.Ud�1; d � 1/Q ! � � � ! CH0.U0; 0/Q ! 0:

The maps CH0.Ui ; i/Q ! CH0.Xi�1; i � 1/Q ! CH0.Ui�1; i � 1/Q arisefrom the localization sequence.

Proof. a) follows because the spectral sequence (1) collapses, and b) by a diagramchase.

Remark. If we fix a smooth schemeX of dimensiond , and use cohomological notation,then by Proposition 2.1 and the Gersten resolution, we get that the rational motiviccomplex Q.d/ is conjecturally concentrated in degree d , say C d D Hd .Q.d// DCH0.�; d /Q D HW

d.�;Q/. Then Conjecture L.0/ says that H i .U; C d / D 0 for

U X affine and i > 0. This is analog to the mod p situation, where the motiviccomplex agrees with the logarithmic de Rham–Witt sheaf Z=p.n/ Š �d Œ�d�, andH i .Uet; �

d / D 0 for U X affine and i > 0. The latter can be proved by writing�d as the kernel of a map of coherent sheaves and using the vanishing of cohomologyof coherent sheaves on affine schemes. This suggest that one might try to do the samefor C d .

4.1 Frobenius action. Let F W X ! X be the Frobenius morphism induced by theqth power map on the structure sheaf.

Theorem 4.6. The push-forward F� acts like qn on CHn.X; i/, and the pull-back F �acts on H i .X;Z.n// as qn for all n.

The theorem is well known, but we could not find a proof in the literature. Theproof of Soulé [14, Proposition 2] for Chow groups does not carry over to higherChow groups, because the Frobenius does not act on the simplices �n, hence a cycleZ �n �X is not send to a multiple of itself by the Frobenius. We give an argumentdue to M. Levine.

Proof. Let DM� be Voevodsky’s derived category of bounded above complexes ofNisnevich sheaves with transfers with homotopy invariant cohomology sheaves. Then

424 T. Geisser

we have the isomorphisms

CHn.X; i/ Š HomDM�.Z.n/Œ2nC i �;Mc.X//;

H i .X;Z.n// Š HomDM�.M.X/;Z.n/Œi �/:

The action of the Frobenius is given by composition with F W Mc.X/ ! Mc.X/ andF W M.X/!M.X/, respectively.

The Frobenius acts on the category DM�, i.e. for every ˛ 2 HomDM�.X; Y / wehave FY ı ˛ D ˛ ı FX . This follows by considering composition of correspondences.Hence it suffices to calculate the action of the Frobenius on Z.n/, i.e. show thatF D qn 2 HomDM�.Z.n/;Z.n// Š Z. But HomDM�.Z.n/;Z.n// is a directfactor of HomDM�.Z.n/;PnŒ�2n�/ D CHn.Pn/. The latter is the free abelian groupgenerated by the generic point, and the Frobenius acts by qn on it.

Remark. 1) It would be interesting to write down an explicit chain homotopy betweenF� and qn on zn.X;�/.

2) The proposition implies that the groups CHn.Fq; i/ are killed by qn�1, and thatCHn.X; i/ is q-divisible for n < 0.

Granted the theorem, the standard argument gives the following corollary, see alsoJannsen [10, Theorem 12.5.7].

Corollary 4.7. Assume that for all smooth and projectiveX of dimension d , the kernelof the map CHd .X �X/! Endhom.M.X// is nilpotent. Then ConjectureP.0/ holds.

The hypothesis of the corollary is satisfied ifX is finite dimensional in the sense ofKimura [13].

Proof. Using the existence of a zero-cycle c of degree 1, we see that the projector�2d D ŒX � c� is defined. Let zX be the motive ker �2d D X=Ld , where L is theLefschetz motive. Consider the action of the geometric Frobenius F 2 Endrat. zX/ CHd .X � X/. Its image in the category of motives for homological equivalence

is algebraic, and its minimal polynomial P zX .T / has roots of absolute value qj2 for

0 � j < 2d . By hypothesis, P zX .F /a D 0 in Endrat. zX/ for some integer a, but by thetheorem, F � acts like qd on CHd .X; i/. Hence 0 6D P zX .qd /a D P zX .F �/a D 0.

References

[1] S. Bloch, Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), 267–304.

[2] T. Geisser, Tate’s conjecture, algebraic cycles and rational K-theory in characteristic p,K-Theory 13 (1998), 109–122.

[3] T. Geisser, Motivic cohomology, K-theory, and topological cyclic homology, in Handbookof K-theory, Vol. 1, Springer-Verlag, Berlin 2005, 193–234.

[4] T. Geisser, Duality via cycle complexes, Preprint 2006.

[5] T. Geisser,Arithmetic homology and an integral version of Kato’s conjecture, Preprint 2007.


[6] T. Geisser, M. Levine, The p-part of K-theory of fields in characteristic p, Invent. Math.139 (2000), 459–494.

[7] H. Gillet, Homological descent for the K-theory of coherent sheaves, in Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), Lecture Notes in Math.1046, Springer-Verlag, Berlin 1984, 80–103.

[8] H. Gillet, C. Soulé, Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996),127–176.

[9] U. Jannsen, Mixed motives and algebraicK-theory, Lecture Notes in Math. 1400, Springer-Verlag, Berlin 1990.

[10] U. Jannsen, On finite-dimensional motives and Murre’s conjecture, in Algebraic cycles andmotives, Vol. 2, London Math. Soc. Lecture Note Ser. 344, Cambridge University Press,Cambridge 2007, 112–142. finite dimensional motives and Murre’s conjecture, 2006.

[11] U. Jannsen, Hasse principles for higher-dimensional fields, Preprint Universität Regensburg18/2004.

[12] B. Kahn, Some finiteness results for etale cohomology, J. Number Theory 99 (2003), 57–73.

[13] S. Kimura, Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005),173–201.

[14] C. Soulé, Groupes de Chow et K-théorie de variétés sur un corps fini, Math. Ann. 268(1984), 317–345.

[15] V. Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in anycharacteristic, Internat. Math. Res. Not. 2002 (2002), 351–355.

Axioms for the norm residue isomorphism

Charles Weibel�

Introduction

The purpose of this paper is to give an axiomatic framework for proving (by inductionon n) that the norm residue mapKMn .k/=`! Hn

ét .k; �˝n`/ is an isomorphism, for any

prime ` > 2 and for any field k such that ` is invertible in k.Fix ` and n. By a Rost variety for a sequence a D .a1; : : : ; an/ of units in k, we

shall mean a smooth projective variety X of dimension d D `n�1 � 1 satisfying theconditions of [8, 6.3] or [6, (0.1)]; the exact definition is given in Definition 1.1 below.One key requirement is that fa1; : : : ; ang vanishes in KMn .k.X//=`. Rost constructedsuch a variety in his 1998 preprint [3]; the proof that it satisfies these properties waspublished in [1] and [6].

Theorem 0.1. Let n and ` be such that the norm residues maps are isomorphisms forall i < n. Suppose that for every sequence a there is a direct summandM of the motiveof a Rost variety X , satisfying the Axioms given in 0.3 below.

Then the norm residue map KMn .k/=`! Hnét .k; �

˝n`/ is an isomorphism.

To state the axioms, let X denote the simplicial Cech scheme p 7! XpC1; see[10, 9.1]. By abuse of notation, we will regard X as a cochain complex, and hence asan element of DMeff. By [10, 9.2], X˝ X ' X and X ˝ X ' X .

There is a duality isomorphismX�˝Ld Š X , whereX� is the dual Chow motiveof X and L is the Lefschetz motive Z.1/Œ2�; see [2, 20.11]. If M is a summand of the

motive of X , the motivic structure map Xp�! Z induces a relative map y W M ! X;

its X-dual Dy is defined to be the composite

Dy W X˝ Ld ! Ldp�˝Ld��! X� ˝ Ld !M � ˝ Ld : (0.2)

Axioms 0.3. (a) M is a direct summand of X . We write y for M ! X ! X.(b) The evident duality map M � ˝ Ld !M is an isomorphism.(c) There is a motive D, related to y by two distinguished triangles:

D ˝ Lb !My�! X!; (0.4)

X˝ LdDy�! M ! D ! : (0.5)

Here b D d=.` � 1/ D 1C `C � � � C `n�2, and Dy is defined in (0.2) via 0.3 (b).

�Weibel’s research was supported by NSA grant MSPF-04G-184, and by the Oswald Veblen Fund.

428 C. Weibel

Given (0.4) and axiom 0.3 (b), triangle (0.5) is equivalent to the duality assertionthat D� ˝ Ld�b Š D; (0.5) is isomorphic to Ld times the dual of triangle (0.4).

Remark 0.6. In the language of Chow motives, any direct summand M D .X; e/ ofX has a dual motive M � D .X; et ; d / and a transpose summand M 0 D .X; et / of X .Thus the summandM 0 DM �˝Ld of X is a priori different fromM . Axiom 0.3 (b)says thatM 0 ! X !M is an isomorphism. (The differences between these languagesis partly due to the fact that the category of Chow motives embeds contravariantly intoDMeff.)

Although we only use X-duals fleetingly in Lemma 3.5 below, they fit into a largercontext, which we shall now quickly describe. For this, we need to assume that kadmits resolution of singularities, so that A� exists by [2, 20.3].

DMX and X-duals 0.7. Let DMX denote the full subcategory of DM consisting ofobjectsM isomorphic to A˝X for a geometric motive A over k. For this, we assumethat k admits resolution of singularities, so that A� exists by [2, 20.3]. As pointed outin [11, 6.11], Hom.M;B/ Š Hom.M;B ˝ X/ for all B in DMeff. Thus

Hom.U˝X; A�˝X/ D Hom.U˝X; A�/ D Hom.U˝X˝A;Z/ D Hom.U˝M;X/:That is, HomX.M;X/ D A� ˝X is an internal Hom object from M to X in the tensorcategory DMX. As such, HomX.M;X/ is unique up to canonical isomorphism (cf. [11,8.1]). Any map M1 !M2 induces a map HomX.M2;X/! HomX.M1;X/ via

Hom.A1 ˝ X; A2 ˝ X/ Š Hom.A1 ˝ X˝ A�2;X/ Š Hom.A�2 ˝ X; A�1 ˝ X/:

It is easy to check that HomX.�;X/ is a contravariant functor, and that it is a dualityin the sense that HomX.HomX.M;X/;X/ ŠM .

In this paper we consider the X-dual DM.�/ D HomX.�;X/ ˝ Ld , which alsosatisfies D2.M/ Š M . Our choice of the twist is such that D.X/ Š X , D.M/ is thetranspose M 0 of Remark 0.6, and Dy is given by (0.2).

This paper is an attempt to clarify the ending of the Voevodsky–Rost programto prove that the norm residue map is an isomorphism, and specifically the proof aspresented in Voevodsky’s 2003 preprint [8]. One important feature of our Theorem 0.1is that only published results (and [3]) are used.

When ` D 2, triangle (0.4) was constructed in [10, 4.4] using D D X, and (0.5) isits X-dual. Axioms 0.3 (a)–(b) hold by a result of Rost (cited as [10, 4.3]).

When ` > 2, there are two different programs for constructing M . Both start byusing the norm residue symbol of a to construct a nonzero element � of H 2bC1;b.X/.The construction of � is given in [10, p. 95] and [8, (5.2)] (see 2.5 below).

In Voevodsky’s approach, the triangles (0.4) and (0.5) are constructed in (5.5) and(5.6) of [8] with M D M`�1 and D D M`�2, starting with � 2 H 2bC1;b.X/. Theduality axiom 0.3 (b) for M and D is given by [8, 5.7].

Unfortunately, there is a gap in the proof of Lemma [8, 5.15], which asserts thatAxiom 0.3 (a) holds, i. e., that M`�1 is a summand of X . That proof depends via

Axioms for the norm residue isomorphism 429

Theorems 3.8 and 4.4 of [8] upon Lemmas 2.2 and 2.3 of [8], whose proofs are currentlyunavailable. Instead, [8, 5.11] proves that the canonical map X ! X factors throughthe map y W M ! X. Since this paper was written, the gap in the proof was patchedin [12].

In Rost’s approach, � defines an equivalence class of idempotent endomorphismse of X and Rost defines M to be the Chow motive .X; e/. By construction, his Msatisfies Axioms 0.3 (a)–(b). Hopefully it also satisfies Axiom 0.3 (c), the existence oftriangles (0.4) and (0.5).

Notation. The integer n and the prime ` > 2 will be fixed. We will work over a fixedfield k in which ` is invertible. The integer d will always be `n�1�1 and b will alwaysbe d=.` � 1/ D 1C � � � C `n�2.

We fix the sequence of units a D .a1; : : : ; an/, and X will always denote ad -dimensional Rost variety relative to a, satisfying Axioms 0.3.

We will work in the triangulated category of motives DMeff described in [2]. TheLefschetz motive is L D Z.1/Œ2�. Unless explicitly stated otherwise, motivic coho-mology will always be taken with coefficients Z.`/. The notation Hp;q

ét .�/ refers tothe étale motivic cohomology Hp

ét .�;Z.`/.q// defined in [2, 10.1].Finally, I would like to thank the Institute for Advanced Study for allowing me to

present these results in a seminar during the Fall of 2006, and Pierre Deligne for hiscontinued interest and encouragement.

1 Proof of Theorem 0.1

Recall [6, 1.20] that a �n�1-variety over a field k is a smooth projective variety Xof dimension d D `n�1 � 1, with deg sd .X/ 6� 0 .mod `2/. Here sd .X/ is thecharacteristic class of the tangent bundleTX corresponding to the symmetric polynomialPtdj in the Chern roots tj of TX ; see [9, 14.3]. We also recall that H�1;�1.Y / is the

group Hom.Z; Y.1/Œ1�/, and is covariant in Y ; see [2, 14.17].

Definition 1.1. A Rost variety for a sequence a D .a1; : : : ; an/ of units in k is a �n�1-variety such that: fa1; : : : ; ang vanishes in KMn .k.X//=`; for each i < n there is a�i -variety mapping to X ; and the motivic homology sequence

H�1;�1.X2/��

0��

1��! H�1;�1.X/! H�1;�1.k/ .D k�/ (1.2)

is exact. As mentioned in the Introduction, Rost varieties exist for every a by [1], [6].

Example 1.2.1. If n D 1 and L D k.pa1/, then Spec.L/ is a Rost variety for a1,

with the convention that s0.X/ D ŒL W k� D `. When n D 2, the Severi–Brauer varietycorresponding to the degree ` algebra with symbol a is a Rost variety.

Proof of Theorem 0.1. By [10, 6.10], it suffices to prove the assertion

H90.n; `/: HnC1;nét .k/ D 0:

430 C. Weibel

As pointed out in the proof of [10, 7.4], it suffices to show that for every symbola 2 KMn .k/ there is a field extension k � F such that a vanishes in KMn .F /=` andHnC1;nét .k/ ! H

nC1;nét .F / is an injection. When F D k.X/, this is the bottom right

arrow in the flowchart (1.3), which is implicit in [8, 6.11].

H 2C2b`;1Cb`.X/ H 2dC1;dC1.D/ D 03.6

onto

1:4��

HnC1;n.X/ 4:4

exact��

2:4into

��

HnC1;nét .k/ sequence

�� HnC1;nét .k.X//:

(1.3)

The other decorations in (1.3) refer to the properties we need and where they areestablished. Proposition 4.4 states that the bottom row of the flowchart (1.3) is ex-act. By Lemma 2.4 the group HnC1;n.X/ injects (by a cohomology operation) intoH 2C2b`;1Cb`.X/, which is in turn a quotient of H 2dC1;dC1.D/ by 1.4. In Corol-lary 3.6, we prove that H 2dC1;dC1.D/ D 0. All of this implies that HnC1;n

ét .k/ !HnC1;nét .k.X// is an injection, finishing the proof of Theorem 0.1.

One part of the flowchart (1.3) is easy to establish.

Lemma 1.4. The map s W X! D ˝ LbŒ1� in (0.4) induces a surjection:

H 2dC1;dC1.D/ Š H 2b`C2;b`C1.D ˝ LbŒ1�/s�!H 2b`C2;b`C1.X/:

Proof. In the cohomology exact sequence arising from (0.4), the next term isH 2b`C2;b`C1.M/, which is a summand ofH 2b`C2;b`C1.X/. Becauseb`D d C b > d ,this group is zero by the Vanishing Theorem [2, 3.6] – which says that Hn;i .X/ D 0

whenever n > i C dim.X/.

The lower left map in flowchart (1.3) is just the motivic-to-étale map by the followingbasic result, which we quote from [10, 7.3].

Lemma 1.5. The structure map X! Spec.k/ induces isomorphisms

H�;�ét .k/ Š H�;�ét .X/ and H

�;�ét .kIZ=`/ Š H�;�ét .XIZ=`/:

2 Motivic cohomology operations

In the course of the proof, we will need some facts about the motivic cohomologyoperations constructed in [9]. When ` > 2, there are operations P i on H�;�.�IZ=`/of bidegree .2i.`�1/; i.`�1//, for each i � 0 (withP 0 D 1), as well as the Bocksteinoperation ˇ of bidegree .1; 0/. These satisfy the Adem relations given in [5] for theusual topological cohomology operations. In fact, the subring of all (stable) motivic


cohomology operations generated by the P i and ˇ is isomorphic to the topologists’Steenrod algebra A�, developed in [5].

In addition, the motivic operations satisfy the usual Cartan formula P n.xy/ DPP i .x/P n�i .y/, P i .x/ D x` if x 2 H 2i;i .Y IZ=`/, and P i D 0 on Hp;q.Y IZ=`/

when .p; q/ is in the region q � i , p < i C q. These are proven in [9, 9.7–9].Still assuming ` > 2, the dual to the usual Steenrod algebra A� is a graded-

commutative algebra on generators �i in (even) degrees .2ì � 2; ì � 1/ and �i in(odd) degrees .2ì � 1; ì � 1/; see [9, 12.6]. The dual to �i is the motivic cohomologyoperation Qi , which has bidegree .2ì � 1; ì � 1/. Because it is true in A�, the Qiare derivations which form an exterior subalgebra of all (stable) motivic cohomologyoperations. They may be inductively defined by Q0 D ˇ and QiC1 D ŒP ì ;Qi �.

We now quickly establish those portions of [8] that we need, concerning theseoperations on the cohomology of X and the unreduced simplicial suspension †X ofX (the cofiber of X ! cone.X/). The proofs we give are due to Voevodsky, and onlydepend upon [10] and [9]. In particular, they do not depend upon the missing lemmasin op. cit., or upon the Axioms 0.3.

Fix q < n. Because we have assumed that the norm residue map is an isomorphismin weight q, and hence that H90.q; `/ holds, it follows from 1.5 that Hp;q.kIZ=`/ ŠHp;qét .kIZ=`/ Š H

p;qét .XIZ=`/ for p � q, and that H qC1;q

ét .XIZ=`/ D 0. Asobserved in [8, 6.6], it then follows from [10, 6.9] that

Hp;q.†XIZ=`/ D 0 when .p; q/ is in the region q < n; p � 1C q: (2.1)

Since X is a ��n�1-variety, Theorem [10, 3.2] translates to:

Theorem 2.2. If i < n, the sequencesQi��! H�;�.†XIZ=`/ Qi��! are exact.

Remark 2.2.1. A slight generalization is stated in [8, 4.3].

Example 2.2.2. For .p; q/ D .n � 2`C 3; n � `C 1/ we have the exact sequence

Hp;q.†XIZ=`/ Q1��! HnC2;n.†XIZ=`/ Q1��! HnC2`C1;nC`�1.†XIZ=`/:Because p � q � n, the left group is zero by (2.1). Thus the right map Q1 is aninjection on HnC2;n.†XIZ=`/.Lemma 2.3. The motivic cohomology groups H�;�.†X/ have exponent `.

Proof. We may assume, by the usual transfer argument, that k has no extensions ofdegree prime to `. In this case, the variety X has a k0-point for the field k0 D k.pa1/of degree ` over k, by [6, 1.23]. It follows that X˝ Spec k0 ŠM.Spec k0/, and hencethat †X ˝ Spec.k0/ Š 0. Since the composition †X ! †X ˝ Spec.k0/ ! †X ismultiplication by ` (by the usual transfer argument), the result follows.

For any p > q we have Hp;q.Spec k/ D 0 and hence Hp;q.X/ Š HpC1;q.†X/.As a consequence of Lemma 2.3 we have that Hp;q.X/! Hp;q.XIZ=`/ is an injec-tion. By [10, 7.2], the cohomology operations Qi preserve integral classes, so theyinduce integral operations on Hp;q.X/ in this range.

432 C. Weibel

Consider the cohomology operation Q D Qn�1 : : :Q2Q1.

Lemma 2.4. The operation Q W HnC1;n.XIZ=`/ ! H 2C2b`;1Cb`.XIZ=`/ is aninjection, and induces an injection

HnC1;n.X/ ,! H 2C2b`;1Cb`.X/:

Proof (Voevodsky). By the above remarks, it suffices to show that the operation Qfrom HnC2;n.†XIZ=`/ to H 3C2b`;1Cb`.†XIZ=`/ is injective. As illustrated in Ex-ample 2.2.2, it is easy to see from 2.2 and (2.1) that each Qi is injective on thegroup H�;�.†XIZ=`/ containing Qi�1 : : :Q1HnC2;n.†XIZ=`/, because the pre-ceding term in 2.2 is zero.

Remark 2.5. The same argument, given in [8, 6.7], shows that Q0 D Qn�2 : : :Q0 isan injection from Hn;n�1.XIZ=`/ to H 2bC1;b.X/ � H 2bC1;b.XIZ=`/.

If fa1; : : : ; ang ¤ 0 inKMn .k/=`, Voevodsky shows in [8, 6.5] that its norm residuesymbol in Hn

ét .k; �˝n`/ lifts to a nonzero element ı 2 Hn;n�1.XIZ=`/. Using injec-

tivity of Q0, we get a nonzero symbol � D Q0.ı/ 2 H 2bC1;b.X/. This symbol is thestarting point of the construction of the motive M in both the program of Voevodsky[8] and that of Rost [4].

3 Motivic homology

In this section, we prove Corollary 3.6, which depends uponAxiom 0.3 (b) and exactnessof (1.2) via results about the motivic homology of X and M .

We will make repeated use of the following basic lemma. In this section, the notationH�p;�q.Y / refers to the group Hom.Z; Y.q/Œp�/; see [2, 14.17].

Lemma 3.1. For every smooth (simplicial) Y and p > q, Hom.Z; Y.q/Œp�/ D 0.

Proof. We have Hom.Z; Y.q/Œp�/ Š Hp�qzar .k; C�.Y�Gq

m// D Hp�qC�.Y�Gqm/.k/;

see [2, 14.16]. The chain complex C�.Y �Gqm/ is zero in positive cohomological de-

grees, so the Hp�q group vanishes.

Lemma 3.2. The structural map H�1;�1.X/! H�1;�1.k/ D k� is injective.

Proof (Voevodsky). By Lemma 3.1, Hom.Z; Xp.1/Œn�/ D 0 for all n � 2 and all p.Therefore the right half-plane homological spectral sequence

E1p;q D Hom.Z; XpC1.1/Œ�q�/) Hom.Z;X.1/Œp � q�/has no nonzero rows below q D �1, and the row q D �1 yields the exact sequence

0 H�1;�1.X/ H�1;�1.X/ H�1;�1.X �X/:Since (1.2) is exact, this implies the result.


Corollary 3.3. The structural map H�1;�1.M/! H�1;�1.k/ D k� is injective.

Proof. By (0.4) and Lemma 3.2, it suffices to show that Hom.Z;D.bC 1/Œ2bC 1�/D 0.By (0.5) this results from the vanishing of both Hom.Z;M.b C 1/Œ2b C 1�/ andHom.Z;X˝ LbCdC1/ which follows from Lemma 3.1.

Lemma 3.4. H 0;1.X/ D H 2;1.X/ D 0 and H 1;1.XIZ/ Š H 1;1.Spec kIZ/ Š k�.

Proof. The spectral sequence Ep;q1 D H q.XpC1IZ.1// ) HpCq;1.XIZ/ degener-ates, all rows vanishing except for q D 1 and q D 2, because Z.1/ Š O�Œ�1�; see[2, 4.2]. We compare this with the spectral sequence converging to HpCq.XIGm/;H q

zar.Y;O�/ ! H

qét.�;Gm/ is an isomorphism for q D 0; 1 (and an injection for

q D 2). Hence for q � 2 we have H q;1.X/ D H q;1ét .X/ D H q;1

ét .k/ D H q�1ét .k;Gm/.

Remark. The proof of 3.4 also shows thatH 3;1.X/ injects intoH 2ét.k;Gm/ D Br.k/.

Lemma 3.5. H 2dC1;dC1.DIZ/ is the kernel of H�1;�1.M/y�!H�1;�1.k/ D k�.

Proof. From (0.5) we get an exact sequence with coefficients Z:

H 2d;dC1.X˝Ld /! H 2dC1;dC1.D/! H 2dC1;dC1.M/Dy�!H 2dC1;dC1.X˝Ld /:

The first group isH 0;1.X/, which is zero by 3.4, so it suffices to show that the mapDyidentifies with the structural map y. This follows from Axiom 0.3 (b), because for anyu in H�1;�1.M/ D Hom.Z;M.1/Œ1�/, X tensored with the composite

X˝ LdDy�!M � ˝ Ld

u�

�! Z.1/Œ1�˝ Ld D Z.d C 1/Œ2d C 1�

is the X-dual of X.�1/Œ�1� u�!My�! X.

Corollary 3.6. H 2dC1;dC1.D/ D H 2dC1;dC1.DIZ/˝ Z.`/ D 0.

4 Between motivic and étale cohomology

Definition 4.1 ([10, p. 90]). Let L.n/ denote the truncation ��nC1Zét.`/.n/ of the com-

plex in DMeff representing étale motivic cohomology; i. e., Hp.�; L.n// Š Hp;nét .�/

for p � nC 1. Let K.n/ denote the mapping cone of the canonical map Z.`/.n/ !L.n/, and consider the triangle Z.`/.n/! L.n/! K.n/! Z.`/.n/Œ1�.

Lemma 4.2. The map HnC1.X; K.n//y�!HnC1.M;K.n// is an injection.

Proof. By triangle (0.4) we have an exact sequence:

Hn.D ˝ Lb; K.n//! HnC1.X; K.n//y�!HnC1.M;K.n//:

434 C. Weibel

We need to see that the left term vanishes. By (0.5), it is sufficient to show thatHn.M ˝Lb; K.n// andHn�1.X˝Lb`; K.n// vanish. This follows from [10, 6.12],which says that H�.Y.1/;K.n// D 0 for every smooth Y , the assumption that Mis a summand of X , and the consequent collapsing of the spectral sequence Ep;q1 DH q.Xp; K.n//) HpCq.X; K.n//.

Corollary 4.3. The map HnC1.X; K.n//! HnC1.k.X/;K.n// is an injection.

Proof. The map HnC1.X;K.n// ! HnC1.k.X/;K.n// is an injection by [2, 11.1,13.8, 13.10], or by [10, 7.4]. The corollary follows from Lemma 4.2, sinceH�.X;�/!H�.M;�/ is a summand of H�.X;�/! H�.X;�/.Proposition 4.4. There is an exact sequence

HnC1;n.X/! HnC1;nét .k/! H

nC1;nét .k.X//:

Proof. By 1.5, X ! Spec.k/ is an isomorphism on étale motivic cohomology, so wehave HnC1;n

ét .k/ Š HnC1.X; L.n//.The map Spec k.X/! X induces a commutative diagram with exact rows:

HnC1;n.X/ ��

��

HnC1.X; L.n// ��

��

HnC1.X; K.n//

4:3 into��

0 = HnC1;n.k.X// �� HnC1.k.X/; L.n// �� HnC1.k.X/;K.n//:

By 4.3, the right vertical map is an injection. The proposition now follows by a diagramchase.

Acknowledgements. This paper is an attempt to clarify Voevodsky’s preprint [8]. It isdirectly inspired by the work of Markus Rost, especially [4], who explicitly suggestedthe line of attack, based upon Axiom 0.3 (c), which is presented in this paper. I amgreatly indebted to both of them.

References

[1] C. Haesemeyer and C. Weibel, Norm Varieties and the Chain Lemma (after Markus Rost),Preprint 2008, available at http://www.math.uiuc.edu/0900.

[2] C. Mazza,V.Voevodsky and C. Weibel, Lecture notes on motivic cohomology, Clay Monogr.Math. 2, Amer. Math. Soc. , Providence, R.I., 2006.

[3] M. Rost, Chain lemma for splitting fields of symbols, Preprint 1998, available athttp://www.math.uni-bielefeld.de/~rost/chain-lemma.html

[4] M. Rost, On the Basic Correspondence of a Splitting Variety, Preprint 2006, available athttp://www.math.uni-bielefeld.de/~rost

[5] N. Steenrod and D. Epstein, Cohomology Operations, Ann. of Math. Stud. 50, PrincetonUniversity Press, Princeton, N.J., 1962.


[6] A. Suslin and S. Joukhovitski, Norm Varieties, J. Pure Appl. Alg. 206 (2006), 245–276.

[7] A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finitecoefficients, in The arithmetic and geometry of algebraic cycles (Banff, 1998), NATO Sci.Ser. C Math. Phys. Sci. 548, Kluwer, Dordrecht 2000, 117–189 .

[8] V. Voevodsky, On Motivic Cohomology with Z/l coefficients, Preprint 2003, available athttp://www.math.uiuc. edu/K-theory/0639/.

[9] V. Voevodsky, Reduced Power operations in Motivic Cohomology, Inst. Hautes Études Sci.Publ. Math. 98 (2003), 1–57.

[10] V. Voevodsky, Motivic Cohomology with Z=2 coefficients, Inst. Hautes Études Sci. Publ.Math. 98 (2003), 59–104.

[11] V. Voevodsky, Motives over simplicial schemes, Preprint, available at http://www.math.uiuc. edu/K-theory/0638/, 2003.

[12] C. Weibel, The Norm Residue Isomorphism Theorem, Preprint 2007, available athttp://www.math.uiuc.edu/0844.

List of contributors

Arthur Bartels, Mathematisches Institut, Westfälische Wilhelms-Universität Münster,Einsteinstraße 62, 48149 Münster, [email protected]

Paul Bressler, Institute forAdvanced Study, Einstein Drive, Princeton, NJ 08540, [email protected]

Ulrich Bunke, NWF I – Mathematik, Universität Regensburg, 93040 Regensburg,[email protected]

Paulo Carrillo Rouse, Projet d’algèbres d’opérateurs, Université de Paris 7, 175, rue deChevaleret, 75013 Paris, [email protected]

Joachim Cuntz, Mathematisches Institut, Westfälische Wilhelms-Universität Münster,Einsteinstraße 62, 48149 Münster, [email protected]

Siegfried Echterhoff, Mathematisches Institut,WestfälischeWilhelms-Universität Mün-ster, Einsteinstraße 62, 48149 Münster, [email protected]

Grigory Garkusha, Department of Mathematics, University of Wales Swansea, Single-ton Park, Swansea SA2 8PP, [email protected]

Thomas Geisser, Department of Mathematics, University of Southern California, 3620S Vermont Av. KAP 108, Los Angeles, CA 90089, [email protected]

Alexander Gorokhovsky, Department of Mathematics, University of Colorado, UCB395, Boulder, CO 80309-0395, [email protected]

Max Karoubi, Université Paris 7, 2 place Jussieu, 75251 Paris, [email protected]

Wolfgang Lück, Mathematisches Institut, Westfälische Wilhelms-Universität Münster,Einsteinstraße 62, 48149 Münster, [email protected]

Ralf Meyer, Mathematisches Institut, Georg-August-Universität Göttingen, Bunsen-straße 3–5, 37073 Göttingen, [email protected]

438 List of contributors

Fernando Muro, Universitat de Barcelona, Departament d’Àlgebra i Geometria, GranVia de les Corts Catalanes 585, 08007 Barcelona, [email protected]

Ryszard Nest, Department of Mathematics, Copenhagen University, Universitetspar-ken 5, 2100 Copenhagen, [email protected]

Mike Prest, School of Mathematics, University of Manchester, Oxford Road, Manch-ester M13 9PL, [email protected]

Thomas Schick, Mathematisches Institut, Georg-August-Universität Göttingen, Bun-senstraße 3–5, 37073 Göttingen, [email protected]

Markus Spitzweck, Mathematisches Institut, Georg-August-Universität Göttingen,Bunsenstraße 3–5, 37073 Göttingen, [email protected]

Andreas Thom, Mathematisches Institut, Georg-August-Universität Göttingen, Bun-senstraße 3–5, 37073 Göttingen, [email protected]

Andrew Tonks, London Metropolitan University, 166–220 Holloway Road, London N78DB, [email protected]

Boris Tsygan, Department of Mathematics, Northwestern University, Evanston, IL60208-2730, [email protected]

Christian Voigt, Institute for Mathematical Sciences, University of Copenhagen, Uni-versitetsparken 5, 2100 Copenhagen, Denmark; Mathematisches Institut, WestfälischeWilhelms-Universität Münster, Einsteinstraße 62, 48149 Münster, [email protected]

Charles Weibel, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540;Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway,NJ 08854, [email protected]

Wend Werner, Fachbereich Mathematik und Informatik, Westfälische Wilhelms-Uni-versität Münster, Einsteinstraße 62, 48149 Münster, [email protected]

List of participants

Natalia Abad Santos, Universidad Nacional del Sur, Bahía BlancaAlexander Alldridge, Universität PaderbornMarian Anton, University of KentuckyPaul Balmer, ETH, ZürichGrzegroz Banaszak, Posnan UniversityMiquel Brustenga, Universtitat Autónoma de BarcelonaUlrich Bunke, Universität GöttingenRubén Burga, Instituto de Matemática y Ciencias Afines, LimaLeandro Cagliero, Universidad Nacional de CórdobaPaulo Carrillo Rouse, Institut JussieuGuillermo Cortiñas, Universidad de Valladolid & Universidad de Buenos AiresJoachim Cuntz, Universität MünsterEugenia Ellis, Universidad de ValladolidMatthias Fuchs, Facultat des Sciences de LuminyImmaculada Gálvez, London Metropolitan UniversityGrigory Garkusha, University of ManchesterThomas Geisser, University of Southern CaliforniaStefan Gille, ETH, ZürichVictor Ginzburg, University of ChicagoAlexander Gorokhovsky, University of Colorado, BoulderJoachim Grünewald, Universität MünsterJuan José Guccione, Universidad de Buenos AiresDaniel Hernández Ruipérez, Universidad de SalamancaJohn Hunton, University of LeicesterTroels Johansen, Universität PaderbornBernhard Keller, Institut JussieuJoachim Kock, Universtitat Autónoma de BarcelonaMaxim Kontsevich, Institut de Hautes Études ScientifiquesMarc Levine, Northeastern University, ChicagoCatherine Llewellyn-Jones, University of LeicesterAna Cristina López Martín, Universidad de SalamancaWolfgang Lück, Universität MünsterRalf Meyer, Universität GöttingenFernando Muro, Max Planck Institut, Bonn

440 List of participants

Amnon Neeman, Canberra UniversityRyszard Nest, University of CopenhagenPere Pascual, Universitat Politécnica de CatalunyaLlorenç Rubió, Universitat Politécnica de CatalunyaSelene Sánchez Florez, Université de Montpellier 2Darío Sánchez Gómez, Universidad de SalamancaFernando Sancho de Salas, Universidad de SalamancaPaolo Stellari, Università degli Studi di MilanoPatrick Szuta, University of Illinois at Urbana-ChampaignGonçalo Tabuada, Institut JussieuAndreas Thom, Universität GöttingenAndrew Tonks, London Metropolitan UniversityBoris Tsygan, Northwestern UniversityChristian Voigt, Universität MünsterWend Werner, Universität MünsterMariusz Wodzicki, California State University, Berkeley