Università degli Studi di Milano

Universität Duisburg-Essen

Master’s Thesis

presented by

Raffaele M. CarboneID 3039435

The resolution of the diagonaland the derived category on some homogeneous spaces

Advisor

Prof. Dr. Marc N. Levine

Academic year 2015/2016

Fakültat für MathematikUniversität Duisburg-EssenThea-Leymann-Str. 945141 Essen

Contents

0 Introduction 2

1 Preliminaries 51.1 Overview on derived categories and derived functors . . . . . . . . . . . . . . 51.2 Families of generators and exceptional sequences . . . . . . . . . . . . . . . . 71.3 The Fourier-Mukai transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 The Koszul resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Schur functors and tensor products . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Derived category on Pn 152.1 The Beilinson resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 The derived category Db(Pn) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Derived category on the Grassmanian 223.1 Sheaf cohomology on the Grassmannian . . . . . . . . . . . . . . . . . . . . . 223.2 The resolution of the diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 The derived category Db(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Derived category on the Quadric 324.1 Quadrics and Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 The Generalized Koszul resolution . . . . . . . . . . . . . . . . . . . . . . . . 334.3 The resolution of the diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 A special truncated resolution for the diagonal . . . . . . . . . . . . . . . . . 40

4.4.1 More on Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . 414.4.2 Spinor bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.5 The derived category Db(Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Bibliography 47

1

Chapter 0

Introduction

The concept of derived category was introduced by J.-L. Verdier in his work for the thésede doctorat d’État ([Ver96], also appeared in SGA 41/2) done under the supervision of A.Grothendieck. The idea was due to Groethendieck, who had already sketched it in his talks toSéminaire Bourbaki in 1957 ([Gro57b]) and to the International Congress of Mathematiciansin 1958 ([Gro58]), as part of a larger theory about duality for coherent sheaves over schemesextending classical Serre’s duality. Verdier filled in the details of his advisors’ intuition,proposing a suitable formalism of triangulated categories and making explicit the conditionsunder which the construction of derived category was possible.

Roughly speaking, for any Abelian category A, under certain conditions it is possibleto consider the “derived category” D(A) whose objects are chain complexes of objects ofA. Morphisms in the derived category are the “localization” of the usual morphism of chaincomplexes with respect to quasi-isomorphism. For any additive right exact functor A → Bbetween abelian categories, under reasonable conditions there exists a “right derived functor”RF : D(A) → D(B) such that, if an object X ∈ A is regarded as a complex concentratedin degree zero, then the ith cohomology group of RF (X), H i(RF (X)), coincides with theusual ith right derived functor of F evaluated at X, RiF (X). Dually, left derived functorsarise with similar properties.

The interesting case of the construction of the derived category associated to an Abeliancategory was to the categories Coh(X) and QCoh(X) of quasi-coherent and coherent sheavesover a given algebraic variety.

Recall that, if OX is the sheaf of regular functions over an algebraic variety (or moregenerally a scheme) X, a sheaf of OX -module is called quasi-coherent if locally it can be rep-resented as the cokernel of a homomorphism of free sheaves; if these free sheaves are of finiterank, the sheaf is called coherent. To any algebraic variety X, there are the correspondingcategories Coh(X) and QCoh(X) of coherent and quasi-coherent sheaves over X.

It was Groethendick himself that, in his Tohoku paper, had presented the concept ofAbelian category in order to describe the properties of Coh(X) and QCoh(X) in analogywith those of the well-known category of modules over a given ring ([Gro57a]). The passagefrom Abelian categories to Derived categories became then necessary in order to solve somedifficulties that had arisen in the study of natural functors, like inverse image and directimage of sheaves, that are not exact.

One on the first important results made possible by the introduction of the new frameworkwas the proof of the Duality Theorem for cohomology of quasi-coherent sheaves, with respectto proper morphisms of locally noetherian schemes, presented by R. Hartshorne in his lecturenotes about Residues and Duality ([Har66]).

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In the years that followed, efforts were made to give a description of the derived categoryof coherent sheaves over a number of algebraic varieties.

A basic question, dealing with the description of a given triangulated category, is theexistence of a suitable family of generators, i.e. a family of objects whose smallest full trian-gulated category containing them is equivalent to the source category. Since a triangulatedcategory is essentially an additive category endowed with a cone construction and a transla-tion functor satisfying a certain number of axioms, the “smallest full triangulated subcategorycontaining a certain family of objects” is simply the full subcategory whose objects can beobtained by taking arbitrary cones, sums and translations of the objects in the family andarrows between them. Families of generators with certain orthogonality properties, calledcomplete (or full) exceptional sequences, are also of particular interest since they allow adescription of the triangulated category in terms of certain homotopy categories.

The first steps in this direction were made by A. A. Beilinson and ([Bei78]) and I. N.Bernshtein, I. M. Gel’fand, and S. I. Gel’fand ([BGS78]), who described the derived categoryof coherent sheaves over the projective space providing two distinguished families of genera-tors. In particular, the idea used by Beilinson – the celebrated “resolution of the diagonal” –opened up to the computation of distinguished families of generators also in the case of ho-mogeneous spaces like Grassmannians, flag varieties, quadrics, and incidence varieties. Thisdevelopment was carried on by M. Kapranov in a series of paper appeared between 1985 and1988 ([Kap85], [Kap86], [Kap88]).

In the present work, we give an account of the results of Beilinson ([Bei78]) and Kapranov([Kap85], [Kap86], [Kap88]) about the derived categories of bounded complexes of coherentsheaves over the projective space, Grassmannians and quadrics.

In Chapter 1, we fix the notations and recall some useful construction and results thatwill appear frequently in the course of the following chapters. In particular, we introducethe concept of exceptional sequence for triangulated categories and we show some usefulequivalence of categories arising from complete strong exceptional sequences. We define alsoFourier-Mukai functors between derived categories of coherent sheaves and we prove somebasic results. Finally, a brief introduction to Young diagrams and Schur functors is given.

In Chapter 2, following mainly [Bei78] and [Cal05], we exhibit two distinguished familiesof generators (actually, two complete strong exceptional sequences) for the derived categoriesof bounded complexes of coherent sheaves on the projective space over a fixed field. The con-struction of a family of generators arising from the resolution of the diagonal is a constructionused also in the following chapters.

In Chapter 3, the case of Grassmannians is considered. To fill in the details of the work ofKapranov ([Kap85]), we work out some results for sheaf cohomology over the Grassmannianvariety that are used to find a resolution for the diagonal. In particular, we provide anexplicit computation of the global sections of the hyperplane bundle and of the quotientbundle. Finally, a description of the derived category on the Grassmannian is provided.

In Chapter 4, we consider the case of non-degenerate quadrics. The main effort in this caseis made for finding a suitable bounded resolution for the diagonal, using widely the Cliffordalgebra associated to the quadric. We mainly follow [Kap86] and [Kap88, §4]. In §1, we addthe proof of the existence of a “generalized Koszul resolution” of the base field, regarded as amodule over the coordinate algebra of the quadric. In its generality, this result is related tothe “Priddy duality” between the coordinate algebra and the Clifford algebra, that we touchupon briefly. In sections 2 and 3 we fill in the details in Kapranov’s construction, and insection 4 we state the final result.

3

Aknowledgements. I wish to thank especially my advisor, Professor M. N. Levine, forsuggesting me this topic and for the constant supervision and help, especially in the lastmonth. I deeply thanks Dr. A. Chatzistamatiou for helpful comments and proofreading. Iam also thankful to L. Mantovani for giving me hints and support in some crucial momentsduring the last semester.

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Chapter 1

Preliminaries

1.1 Overview on derived categories and derived functors

In the present work, general facts about triangulated and derived categories will be consid-ered well known. Standard references for this material are the Lecture notes by R. Hartshorneabout Residues and Dualiy ([Har66]) and the book by C. Weibel ([Wei94, Chapter 10]). Inthis section we fix the notations and we recall the construction of derived categories andderived functors of particular interest for our discussion.

Let A an abelian category. We recall that the homotopy category of chain complexes of A,namely K(A), is a triangulated category whose objects are chain complexes in A and whosemorphisms are chain maps modulo the relation of homotopy (see [Wei94, Exercise 1.4.5]). Thederived category D(A) is obtained by localization of K(A) at the multiplicative set consistingof quasi-isomorphims and is endowed with a localization functor q : K(A)→ D(A), which isa functor of triangulated categories (i.e. preserves exact triangles and translations).

Moreover, special subcategories of K(A) can be considered. In particular, K+(A),K−(A),Kb(A) denote the homotopy category of bounded below, bounded above and boundedchain complexes. The corresponding derived categories are denoted with D+(A), D−(A),Db(A).

If X is an algebraic variety over a field (or, more generally, a scheme) with sheaf of regularfunctions OX , it is well known that the categories Coh(X) and QCoh(X) of coherent andquasi-coherent sheaves over X are full thick subcategories of OX -Mod, the Abelian categoryof sheaves of OX -modules ([Har77, Chapter II.5]).

Definition 1.1.1. If X is a scheme, we define the homotopy category on X, denoted byKb(X), to be the homotopy category Kb(Coh(X)) of bounded chain complexes of coherentsheaves over X. We define the derived category on X, denoted by Db(X), to be thederived category Db(Coh(X)) corresponding to Kb(X).

Derived functors will be also considered the present work. Let F : A → B be an additivefunctor between Abelian categories. Let Kb(A), Kb(B) the bounded homotopy categoriesand let us denote with F : Kb(A) → Kb(B) the induced functor at the level of homotopy.Let Db(A) and Db(B) be the bounded derived categories of A and B respectively and denotewith qA : Kb(A)→ Db(A) and qB : Kb(B)→ Db(B) the localization functors.

Definition 1.1.2. A (total) right derived functor of F on Kb(A) is a functor of triangu-lated categories RF : Db(A)→ Db(B), together with a natural transformation ξ from qB F

5

to (RF ) qA which is universal, in the following sense: if G : Db(A) → Db(B) is anotherfunctor of triangulated categories endowed with a natural transformation ζ : qBF

.−→ GqA,then there exists a unique natural transformation η : RF .−→ G such that ζA = ηqAA ξA forany object A in A.

A (total) left derived functor of F onKb(A) is a functor of triangulated categories LF :Db(A) → Db(B), together with a natural transformation ξ from (LF )qA to qBF satisfyingthe dual universal property.

In general, proving the existence of a left or right derived functor of a given additivefunctor is not trivial. Without entering in the details of such proofs, we give a brief summaryof some standard results about derived functors between categories of sheaves, for schemesX over a base field k, that will be used in the further discussion. All results presented in thisparagraph are covered by [Huy06, Chapter 3.3].

• Suppose that X is projective, noetherian. The global section functor Γ : Coh(X) →k-mod admits a right derived functor RΓ : Db(X) → Db(k-mod). If F is a coherentsheaf on X, for any i ≥ 0 the usual cohomology groups H i(X,F) are equal to thegroups H i(RΓ(F)), where F is regarded as a complex concentrated in degree 0.

• The tensor product of coherent sheaves over X can be regarded as a bifunctor

_⊗_ : Coh(X)× Coh(X)→ Coh(X).

In the case X is smooth, it admits a left derived (bi)functor:

_⊗L _ : Db(X)×Db(X)→ Db(X).

• If X → Y is a morphism of schemes, the inverse image sheaf functor f∗ and the directimage sheaf functor f∗ can be considered.

Under the condition that f is projective (or proper), the direct image functor f∗ restrictsto a functor Coh(X) → Coh(Y ) and admits a right derived functor Rf∗ : Db(X) →Db(Y ). If f and g are composable proper morphisms, there is a natural canonicalisomorphism

R(g f)∗ ' Rg∗ Rf∗.

Deriving the inverse image sheaf f∗ could be in general more complicated. In general,f∗ is right exact, so a left derived functor Lf∗ can be defined. In our discussion, wewill work mainly with flat morphisms, in which case the inverse image functor f∗ turnsout to be exact. In consequence, f∗ need not to be derived to get a functor of derivedcategories f∗ : Db(Y )→ Db(X).

• We need also to consider compatibility properties between these derived functors.

i) If f : X → Y is a proper morphism between projective schemes, for any pairof objects F in Db(X), G in Db(Y ), there is a natural isomorphism (projectionformula):

Rf∗(F)⊗L G ' Rf∗(F ⊗L Lf∗(G)) (1.1)

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ii) If f : X → Y is any morphism between projective schemes and F ,G are objectsin Db(Y ), then there is a natural isomorphism:

Lf∗F ⊗L Lf∗G ' Lf∗(F ⊗L G) (1.2)

iii) Suppose to have a fibre product diagram:

X ×Z Y Y

X Z

v

g f

u

with f proper, u flat. For any object F in Db(Y ), there is a natural isomorphism(flat base change):

u∗Rf∗F ' Rg∗v∗F . (1.3)

1.2 Families of generators and exceptional sequences

In this section we introduce the concepts of family of generators and complete exceptionalsequence for a triangulated category; they will be used to give a description of triangulatedcategories such as the derived category on a scheme.

Let C a triangulated category. For any object C in C, we denote with C[i] the image ofC through the translation functor applied i times. Exact triangles will be denoted with thefollowing notation: if (A, B, C) is a triple of objects in C and (u, v, w) is a triple of morphismsu : A→ B, v : B → C, w : C → A[1], we will just write:

Au−→ B

v−→ Cw−→ A[1]

to denote that the triangle is exact.

We will also consider sets of generators for triangulated categories, in the following sense:

Definition 1.2.1. Let C any triangulated category in the sense of Verdier. A family ofgenerators for C is a collection of elements in C such that the smallest full triangulatedsubcategory containing them is equivalent with C itself.

Remark 1.2.2. Let X = XiI be a family of objects in a triangulated category C. Then,the smallest full subcategory containing the family X (or the “generated subcategory” ofX) can be obtained as the full subcategory whose objects belong to the smallest collectionsatisfying the following properties:

• if Xi is in the collection and i > 0, the translation X[i] is in the collection;

• if Xi, Xj are in the collection, the direct sum Xi ⊕Xj is in the collection;

• if u is a morphism between two elements in the collection, also the cone of u is in thecollection.

In other words, the objects in the generated subcategory of X are those that can be obtainedby taking iteratively translations, direct sums and cones of objects in X and morphismsbetween them.

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Suppose that a triangulated category C is k-linear for some base field k, i.e. the Hom setsin C have a structure of k-vector spaces. In this case, we will be particularly interested infamilies of generators with the distinctive property of being “exceptional”.

Definition 1.2.3. An object E in a k-linear triangulated category C is called exceptionalobject if

Hom(E,E[m]) =

k if m = 0

0 if m 6= 0

Definition 1.2.4. An n-tuple (E1, . . . , En) of exceptional objects in C is called exceptionalsequence if

Hom(Ei, Ej [m]) = 0 whenever 1 ≤ j < i ≤ n, for all m ∈ Z.

The sequence is called strong exceptional if in addition

Hom(Ei, Ej [m]) = 0 for all 1 ≤ i, j ≤ n, and m 6= 0. (1.4)

A set of n exceptional objects E1, . . . , En that satisfies only Condition 1.4 (the order beingno more relevant) is called simply a strong exceptional set.

Definition 1.2.5. An exceptional sequence (E1, . . . , En) (or an exceptional set E1, . . . , En)is called complete (or full) if the objects Ei’s give a family of generators for C.

We will consider exceptional sequences and sets especially in the case where our trian-gulated category C is the derived category Db(A) arising from an Abelian category A withenough injectives. In this case, strong exceptional sets acquire particular relevance.

Proposition 1.2.6. Let A an abelian category with enough injectives and let E1, . . . , En astrong exceptional set in Db(A), consisting of objects of A regarded as complexes concentratedin degree zero. Denote with Kb(E1, . . . , En) the homotopy category of bounded complexeswhose terms are finite direct sums of the Ei’s. Then, the functor:

ΦE1,...,En : Kb(E1, . . . , En) −→ Db(A)

obtained by the composition of the inclusion i : Kb(E1, . . . , En)→ Kb(A) with the localiza-tion functor qA : Kb(A)→ Db(A) is fully faithful.

We prove the proposition with the aim of the following technical lemmas, whose proof isin its turn inspired by similar lemmas proved in [AO89], but under different conditions.

Lemma 1.2.7. Let K·, L· be chain complexes in A such that K is upper bounded, L is lowerbounded and ExtpA(Ki, Lj) = 0 for any pair of indexes i, j ∈ Z and p > 0. Let f : K → Lany chain map. If L is acyclic, then f is null homotopic.

Proof. Since L is lower bounded, we may assume that Lj = 0 for any j < 0. Similarly, sinceK is upper bounded, we assume that Ki = 0 for any i > n. First, split L into short exactsequences:

0→ Qi−1 → Li → Qi → 0

defined for i ≥ 1, Q0 being equal to L0. Then, using the long exact sequence of cohomology,we obtain by induction that all the terms Qj satisfy Extp(Ki, Qj) = 0 for all p > 0. Now, to

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prove that f is null homotopic we need to construct a chain homotopy s between f and thezero map, i.e. a collection of maps si : Ki → Li−1 such that fi = si+1di +di−1si for all i ∈ Z.Since L and K are bounded, it is enough to construct such maps for i going from 0 to n (allothers being zero). We define such maps arguing by induction starting from i = n. Considerthe diagram:

. . . Kn−1 Kn 0 . . .

. . . Ln−1 Ln Ln+1 . . .

dn−1

fn−1 fn

dn

fn+1

dn−1 dn

Since dnfn = 0, the map fn factors through the kernel Qn−1 of the differential dn : Ln →Ln+1. To define sn, consider the short exact sequence:

0→ Qn−2 → Ln−1 → Qn−1 → 0. (1.5)

Since Ext(Ki, Qn−2) = 0 for any i ∈ Z, we can apply Hom(Kn,−) to get a short exactsequence:

0→ Hom(Kn, Qn−2)→ Hom(Kn, Ln−1)→ Hom(Kn, Qn−1)→ 0.

Choose a lifting of fn ∈ Hom(Kn, Qn−1) and use it to define sn : Kn → Ln−1; by construction,it satisfies dn−1sn = fn. To set up induction, apply now the functor Hom(Kn−1,−) to theshort exact sequence (1.5). By diagram chase, the map fn−1 − sndn−1 : Kn−1 → Ln−1

restricts to a morphism in Hom(Kn−1, Qn−2). Hence, with the aim of the sequence

0→ Qn−3 → Ln−2 → Qn−2 → 0

we can apply the same argument as above to define a morphism sn−1 : Kn−1 → Ln−2 suchthat dn−2sn−1 = fn−1− sndn−1. Iterating this argument a finite number of times, we get thedesired chain homotopy.

Lemma 1.2.8. Let A be an abelian category with enough injectives and let K·, L· be boundedchain complexes of objects in A such that ExtpA(Ki, Lj) = 0 for any pair of indexes i, j ∈ Zand for all p > 0. Then:

HomKb(A)(K,L) ' HomDb(A)(K,L).

Proof. Since A has enough injectives, there exists a quasi isomorphism of chain complexesf : L→ I such that I is a bounded below complex of injective elements. Denote with C(f)the cone of f , which fits into the exact triangle:

L→ I → C(f)→ L[1]. (1.6)

Notice that the cone C(f) is a complex whose jth term is equal to Lj+1 ⊕ Ij . Then:

ExtpA(Ki, Cj(f)) ' ExtpA(Ki, Lj+1)⊕ ExtpA(Ki, Ij) = 0

for any i, j ∈ Z and p > 0. Moreover, f is a quasi-isomorphism, hence the cone C(f) is acylic.Then, the previous lemma shows that any chain map from K to C(f) is homotopyc to zero;or, in other words:

HomK(A)(K,C(f)) = 0.

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The long exact sequence associated to the exact triangle (1.6) hence shows that:

HomK(A)(K,L) ' HomK(A)(K, I).

Now, it is well known that, being I a bounded below complex of injective elements, there isa natural isomorphism:

HomK(A)(K, I) ' HomD(A)(K, I).

On the other hand, L and I are isomorphic in the derived category. Hence, we conclude that:

HomKb(A)(K,L) = HomK(A)(K,L) '

' HomK(A)(K, I) '' HomD(A)(K, I) '' HomD(A)(K,L) = HomDb(A)(K,L).

We are now ready to prove Proposition 1.2.6.

Proof. Let E1, . . . , En be objects in A that gives a strong exceptional set for Db(A). Inparticular, recall that

HomDb(A)(Ei, Ej [m]) = 0 for all 1 ≤ i, j ≤ n, and m 6= 0

Now, the Hom group HomDb(A)(Ei, Ej [m]) coincides with the hyperext group ExtmDb(A)(Ei, Ej),which in its turn coincides with the classical extension ExtmA(Ei, Ej) since Ei and Ej are ele-ments of A which has enough injectives ([Wei94, Corollary 10.7.5]). Let K and L be boundedchain complexes whose terms are finite direct sums of the Ei’s. Since Extp commutes withfinite direct sums in each component, it follows that K and L satisfies the hypothesis of thelast lemma. In particular, this shows that for any pair of elements K,L in Kb(E1, . . . , En)we have:

HomKb(E1,...,En)(K,L) = HomKb(A)(iK, iL) = HomDb(A)(ΦE1,...,En(K),ΦE1,...,En(L))

This means that ΦE1,...,En is fully faithful.

Proposition 1.2.6 gives us the following result, which appears in [Kap88, Lemma 1.6]without proof.

Proposition 1.2.9. Let X a noetherian scheme and let E1, . . . , En a complete strongexceptional set in Db(X) consisting of objects of Coh(X) regarded as complexes concentratdin degree zero. Denote with Kb(E1, . . . , En) the homotopy category of bounded complexesof sheaves whose terms are finite direct sums of the Ei’s. Then, the functor:

ΦE1,...,En : Kb(E1, . . . , En) −→ Db(X)

obtained by the composition of the inclusion i : Kb(E1, . . . , En) → Kb(Coh(X)) with thelocalization functor qA : Kb(Coh(X))→ Db(X) is an equivalence of categories.

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Proof. The category Coh(X) has not enough injectives, so we can not apply directy Propo-sition 1.2.6. Consider then the inclusion:

Coh(X) → QCoh(X)

The corresponding functor at the level of derived categories:

j : Db(X)→ Db(QCoh(X))

defines an equivalence between the derived category Db(X) of X and the full triangulatedsubcategory Db

coh(QCoh(X)) of bounded complexes of quasi-coherent sheaves with coherentcohomology ([Huy06, Proposition 3.5]). In particular, the functor j is fully faithful and hencethe set E1, . . . , En is strong exceptional in Db(QCoh(X)).

Consider now the composition of functors:

Kb(E1, . . . , En)ΦE1,...,En−−−−−−−→ Db(X)

j−→ Db(QCoh(X)).

The composition j ΦE1,...,En coincides with the functor:

ΦQCohE1,...,En : Kb(E1, . . . , En) −→ Db(QCoh(X))

associated to the strong exceptional set E1, . . . , En in Db(QCoh(X)). Since QCoh(X)has enough injectives ([Har66, Theorem 7.18]), the functor ΦQCoh

E1,...,En is fully faithful thanksto Proposition 1.2.6. Since j is fully faithful, we conclude that ΦE1,...,En is fully faithful too.Moreover, ΦE1,...,En is essentially surjective since E1, . . . , En is a complete exceptional setfor Db(A). Then ΦE1,...,En is an equivalence of categories as claimed.

1.3 The Fourier-Mukai transform

We introduce now a functor between derived categories which plays an important rolein the computation of generators for derived categories on schemes. We assume, where notspecified, that all schemes are projective and smooth over a fixed field k.

Definition 1.3.1. Let X,Y smooth projective varieties over a field k and denote with X×Ytheir product, endowed with the canonical projections p = π1 : X×Y → X, q = π2 : X×Y →Y . Let E be an object in Db(X × Y ). Define the induced Fourier-Mukai transform([Huy06, Chapter 5]) to be the functor

ΦEX→Y : Db(X) −→ Db(Y )

ΦEX→Y (−) = Rq∗(p∗(−)⊗L E)

whereRp∗ and ⊗L are the derived direct image and the derived tensor product; the morphismp is flat, so p∗ needs no derivation.

Since all the functors involved are functors between triangulated categories, the compo-sition ΦEX→Y preserves exact triangles. Moreover, also the dependence on E is functorial andpreserves exact triangles, i.e. if E → F → G → E [1] is an exact triangle in Db(X × Y ), thenfor any object A in Db(X) we have an exact triangle

ΦEX→Y (A)→ ΦFX→Y (A)→ ΦGX→Y (A)→ ΦEX→Y (A)[1].

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A notable case is obtained when we consider X = Y , so that p and q are both mapsfrom X ×X to X. The morphism i = (idX , idX) : X → X ×X is an isomorphism onto thediagonal ∆ ⊂ X ×X; the canonical sheaf of ∆ is O∆ = i∗OX . Regarding O∆ as a complexof sheaves concentrated in degree zero, it is an element in Db(X × X). Taking E = O∆ asthe kernel, the induced Fourier-Mukai transform becomes:

ΦO∆X→X =Φ∆ : Db(X) −→ Db(X)

Φ∆(−) = Rq∗(p∗(−)⊗L O∆)

Proposition 1.3.2. For any object F in Db(X), Φ∆(F) ' F naturally.

Proof. First notice that i : X → X×X is a closed immersion, hence the direct image functori∗ is exact and needs no derivation. It follows that O∆ = i∗OX also in the derived categoryDb(X ×X). Then:

Φ∆(F) = Rq∗(p∗(F)⊗L O∆) = Rq∗(p∗(F)⊗L Ri∗OX)

We can apply the projection formula stated above:

Φ∆(F) ' Rq∗(Ri∗(i∗p∗(F)⊗L OX))

The canonical sheaf OX is locally free, hence the tensor product − ⊗ OX is exact and thederived tensor product is naturally isomorphic to the identical functor. Then we can write:

Φ∆(F) ' Rq∗(Ri∗(i∗p∗(F))) '' R(q i)∗((p i)∗(F))) ' F

1.4 The Koszul resolution

The Koszul resolution is a general construction in homological algebra that will appearoften in our discussion. We give a brief account of this algebraic tool following [FL85, ChapterIV, §2].

Definition 1.4.1. Let A any ring, let E a free module of finite rank n over A and let d1 alinear homomorphism E → A. The ith exterior powers of E, for i going from 1 to n, fit intoa chain complex, called the Koszul complex:

0→ ΛnEdn−→ . . .

d3−→ Λ2Ed2−→ Λ1E

d1−→ A→ 0 (1.7)

where the boundary map di is defined by the formula:

di(t1 ∧ · · · ∧ ti) =i∑

j=1

(−)j−1d1(tj)t1 ∧ · · · ∧ tj ∧ · · · ∧ ti.

If we denote by I the image of E in A, then I is an ideal of A and the above chain complexyields the augmented Koszul complex K(I):

0→ ΛnEdn−→ . . .

d3−→ Λ2Ed2−→ Λ1E

d1−→ Ad0−→ A/I → 0

Under certain condition the augmented Koszul complex is exact, and so it gives a freeresolution of the A-module A/I.

12

Lemma 1.4.2. ([FL85, 2.1])

a) Suppose that I admits a set of n generators that gives a regular sequence for the ringA. Then the augmented Koszul complex is exact.

b) Suppose that A is noetherian and local and a set of n elements in the maximal idealgenerates the ideal I. If the augmented Koszul complex is exact, then such set of nelements gives a regular sequence for the ring A.

c) If I is the unit ideal (i.e. d1 is surjective), then the (augmented) Koszul complex isexact.

The construction of the Koszul complex for free modules over a ring can be generalizedto locally free sheaves of modules over a scheme. Let X a scheme and denote by OX itsstructure sheaf. Suppose E is a locally free sheaf of OX -modules of rank n, endowed with amorphism of OX -modules d1 : E → OX . Similarly to the case of modules, we can constructthe Koszul complex:

0→ ΛnE −→dn

. . . −→d3

Λ2E −→d2

Λ1E −→d1

OX → 0

The previous construction will be mainly applied in the following situation. Suppose Eis a locally free sheaf of OX -modules of rank n. Pick a global section s of E over X. If E∨ isthe dual sheaf of E , as defined in [Har77, Exercise 5.1], then s determines an homorphism:

d1 = s∨ : E∨ −→ OX .

If U ⊂ X is an open subset, s∨U is equal to:

s∨U : Hom(E|U ,OX |U ) −→ OX(U)

f 7−→ fU (s|U )

The image of s∨ is a sheaf of ideals I ⊂ OX , which determines a closed subscheme of Xcalled the “zero scheme” of s and denoted as Z(s). We obtain an augmented Koszul complexK(s):

0→ ΛnE∨ −→dn

. . . −→d3

Λ2E∨ −→d2

Λ1E∨ −→d1

OX −→d0

OZ(s) → 0 (1.8)

For any x ∈ X, the stalk of E at x is a free module Ex over the local ring of functions OX,x;the germ sx ∈ Ex, under a fixed isomorphism Ex ' OnX,x, is represented by a sequence ofelements (a1, . . . , an) of OnX,x. Then, an easy computation shows that the image of s∨x inOX,x is exactly the ideal Ix = (a1, . . . , an), and the stalk of the Koszul complex K(s) at xis the Koszul complex of modules that was previously denoted with K(Ix). Note that, ifx 6∈ Z(s), then Ix = 1 and K(Ix) is exact by Proposition 1.4.2, part c. If x ∈ Z(s), by thesame proposition, K(Ix) is exact whenever the sequence a1, . . . , an is a regular sequence forOX,x.

1.5 Schur functors and tensor products

In the last section of this preliminary chapter, we discuss briefly a tool from theory ofrepresentations that turns out to be useful when dealing with tensor products and Grass-mannians. We follow [FH04, Chapter 4] for definitions and classical results. For a concrete

13

description of the space ΣαW introduced below, see also [nLa]. All vector spaces are intendedover a fixed field k.

Let α = (α1, . . . αm) a non-increasing sequence of positive integers. We can represent αas a Young diagram, i.e. as a table with m rows, aligned on the left, such that the ith rowhas exactly αi cells. The size of α, denoted by |α|, is the sum |α| =

∑mi=1 αi. The transposed

diagram α∗ is the one obtained exchanging the rows and columns in α.

Definition 1.5.1. Letm a positive integer and let α = (α1, . . . αm) a non-increasing sequenceof m positive integers. The Schur functor Σα associated to α is defined as a functor:

Σα : k-vector spaces → k-vector spaces

such that, for any vector space W , ΣαW coincides with the image of the Young symmetrizerhα in the space of tensors of W of rank m.

An useful application of the Schur functors comes out in the description of the spaces ofexterior power of a tensor product, as stated in the following lemma ([Kap85, Lemma 0.5]):

Lemma 1.5.2. Let V and W be two vector spaces and let p be a non negative integer. IfΛp(V ⊗W ) denotes the vector space of pth exterior powers, then we have a natural isomor-phism of GL(V )×GL(W )-modules:

Λp(V ⊗W ) ' ⊕|α|=pΣαV ⊗ Σα∗W

where α runs over all Young diagrams with size p and each summand occurs with multiplicityone.

14

Chapter 2

Derived category on Pn

In this chapter we find two distinguished complete strong exceptional sets for the derivedcategory on the projective space over a given field k. The main tool used in this chapter,the resolution of the diagonal, is a celebrated result by Beilinson ([Bei78]). However, in thepresent exposition, we follow mainly [Cal05] and [Cal09].

Let k be a fixed field and let X = Pnk be the projective space, regarded as a projectivescheme over Spec k. On X we consider the structure sheaf OX and the sheaf of differentialsΩX/k ([Har77, II.8]). The product X ×X is endowed with the projections p, q : X ×X → Xrespectively on the first and on the second component. Since X is a smooth, projectivevariety over Spec k, the projections are flat. Hence, for any pair of objects E ,F ∈ Db(X), wecan define:

E F = p∗E ⊗L q∗F ∈ Db(X ×X)

where ⊗L is the derived tensor product in Db(X ×X).

2.1 The Beilinson resolution

Proposition 2.1.1. (Beilinson resolution of the diagonal) With the previous notation, con-sider the diagonal embedding i : X

∼−→ ∆ ⊂ X × X. Then the sheaf O∆ = i∗OX admits alocally free resolution on X ×X:

0→p∗OX(−n)⊗ q∗(Ωn(n))→ p∗OX(−n+ 1)⊗ q∗(Ωn−1(n− 1))→ . . .

· · · → p∗OX(−1)⊗ q∗(Ω(1))→ OX×X → O∆ → 0

Proof. ([Cal09, Theorem III.1.1])We consider, in order to make the computations, three isomorphic copies of the projective

space X = Pnk (the first being X itself), denoted with different variables:

X := Proj(k[x0, . . . , xn])

Y := Proj(k[y0, . . . , yn])

Z := Proj(k[z0, . . . , zn]).

The product X ×X is canonically isomorphic to X × Y , endowed with projections:

p : X × Y → X

q : X × Y → Y

15

On any copy of X we consider the structural sheaves OX , OY , OZ and the sheaf of k-differentials ΩX ,ΩY ,ΩZ . We will compute the resolution of the diagonal as the augmentedKoszul complex (1.8) corresponding to a morphism p∗OX(−1) ⊗ q∗ΩY (1) → OX×Y . Todefine such morphism, consider on any copy of X the Euler exact sequence ([Har77, Theo-rem II.8.13]):

0→ ΩX(1)→ On+1X → OX(1)→ 0 (2.1)

The second morphism of the sequence is sent by p∗ to a morphism:

p∗(On+1X ) −→ p∗(OX(1))

The second morphism of the same sequence on Y is sent by q∗ to a morphism:

q∗(ΩY (1)) −→ q∗(On+1Y )

Then, noting that q∗(OY ) and p∗(OX) are both canonically isomorphic to OX×Y , we canconsider a chain of morphisms:

q∗(ΩY (1)) −→ q∗(On+1Y )

∼−→ (OX×Y )n+1 ∼−→ p∗(On+1X ) −→ p∗(OX(1)) (2.2)

The composition resulting from such chain, tensored by p∗(OX(−1)), yields a morphism:

p∗OX(−1)⊗ q∗ΩY (1) −→ p∗OX(−1)⊗ p∗OX(1)

Finally, considering the chain of natural isomorphisms:

p∗OX(−1)⊗ p∗OX(1) ' p∗(OX(−1)⊗OX(1))

' p∗(OX) ' OX×Y

we finally obtain a morphism:

p∗OX(−1)⊗ q∗ΩY (1) −→ OX×Y (2.3)

Claim. The image of the morphism (2.3) is the sheaf ideal of the diagonal ∆ ⊂ X × Y .

To prove the claim, consider a pair of local charts

U = xi 6= 0 ⊂ XV = yj 6= 0 ⊂ Y

Following [Har77, Theorem II.8.13], recall that OX(U)n+1 is a product of n + 1 copies ofSx := k[x0/xi, . . . , xn/xi]. If we call ek the kth distinguished generator (i.e. the unity elementin the kth copy of Sx), then by the Euler exact sequence ΩX(1)(U) can be identified withthe subring of OX(U)n+1 generated by the elements

ωXr = er − (xr/xi)ei

for r 6= i in 0, . . . , n + 1. On the other side, the second morphism of the Euler exactsequence sends er to xr ∈ OX(1)(U).

The pullback functors p∗ and q∗ restricted to U and V correspond to the pullback to theproduct of affine schemes. If Sx = k[x0/xi, . . . , xn/xi] and Sy = k[y0/yj , . . . , yn/yj ], then p∗|U

16

and q∗|U corresponds to the extension of scalars from Sx and Sy, respectively, to Sx ⊗k Sy.Then the sequence of morphisms (2.2), computed on U ×V , is described by the composition:

ωYr 7−→ er −yryjej 7−→ xr −

yryjxj = xi

(xrxi− yryj

xjxi

)The section OX(−1)(U) is the Sx-module generated by 1/xi, so the morphism in (2.3) isdescribed on a basis by:

1

xi⊗ ωYr 7−→

xrxi− yryj

xjxi∈ OX×Y (U × V )

Denote with R the ideal in OX×Y (U ×V ) generated by the elements xrxi− yr

yj

xjxi

for r 6= j. Toprove that R is equal to the ideal sheaf of the diagonal evaluated at U×V , consider the thirdcopy of X, denoted with Z. The diagonal map Z → X × Y is locally a map W → U × V ,where W = zi 6= 0 and zj 6= 0 ⊂ Z. Then

W = Spec k

[z0

zi, . . . ,

znzi

]zjzi

= Spec k

[z0

zi, . . . ,

znzi,zizj

]and the diagonal map corresponds to the surjective morphism:

OX×Y (U × V ) = k

[x0

xi, . . . ,

xnxi,y0

yj, . . . ,

ynyj

]−→ k

[z0

zi, . . . ,

znzi,zizj

]= OZ(W )

xkxi7−→ zk

ziykyj7−→ zk

zi

zizj

=zkzj

whose kernel is the sheaf ideal of ∆. Direct computation shows that R is contained in suchkernel, so the above map factors through the quotient:

k[x0xi, . . . , xnxi ,

y0

yj, . . . , ynyj

]R

−→ k

[z0

zi, . . . ,

znzi,zizj

]Notice that R contains the relation 1 = xi

xi= yi

yj

xjxi. Hence, we have a natural isomorphism:

k[x0xi, . . . , xnxi ,

y0

yj, . . . , ynyj

]R

' k[x0

xi, . . . ,

xnxi,yiyj

]showing that the sheaf ideal coincides with R. Hence, the claim is proved.

Proved the claim, since p∗OX(−1) ⊗ q∗ΩY (1) is a locally free sheaf of rank n, we canconsider the augmented Koszul complex corresponding to the morphism (2.3):

Λn(p∗OX(−1)⊗ q∗ΩY (1))→ · · · → p∗OX(−1)⊗ q∗ΩY (1)→ OX×Y → O∆

The exactness of this complex is given by the following argument. Suppose to have a pointp ∈ X × Y , lying in U × V . Then, by the proof of the Claim, the stalk Rp is generated by asequence of n elements in OX×Y,p which has dimension 2n. On the other hand, the quotientOX×Y,p/Rp coincides with O∆,p which has dimension n. By Krull’s theorem, it follows that

17

the sequence of germs generating Rp is regular. Then we apply the result of Chapter 1 andconclude that the augmented Koszul complex is exact.

Finally, we apply Lemma 1.5.2 to obtain a canonical isomorphism:

Λk(p∗OX(−1)⊗ q∗Ω(1)) ' p∗OX(−k)⊗ q∗(Ωk(k))

whence the proposition is proved.

2.2 The derived category Db(Pn)

The Beilinson resolution proved in Proposition 2.1.1 allows us to find generators forthe derived category on Pn. First of all, we split the Beilinson resolution into short exactsequences:

0→p∗OX(−n)⊗ q∗(Ωn(n))→ p∗OX(−n+ 1)⊗ q∗(Ωn−1(n− 1))→ Cn−1 → 0

0→Cn−1 → p∗OX(−n+ 2)⊗ q∗(Ωn−2(n− 2))→ Cn−2 → 0

. . .

0→C1 → OX×X → O∆ → 0

Any of such exact sequences can be viewed as an exact sequence of bounded complexes ofcoherent sheaves, concentrated in degree 0. Tensor product by a locally free sheaf and thedirect image functors are exact; hence we can pass to the derived category (i.e. apply thelocalization functor), where the exact sequences are translated into exact triangles:

OX(−n) Ωn(n)→ OX(−n+ 1) Ωn−1(n− 1)→ Cn−1 → (OX(−n) Ωn(n))[1]

Cn−1 → OX(−n+ 2) Ωn−2(n− 2)→ Cn−2 → Cn−1[1]

. . .

C1 → OX×X → O∆ → C1[1]

Recalling the definition of generators for a triangulated category (1.2.1), we conclude thatthe object O∆ belongs to the subcategory of Db(X ×X) generated by the family

OX(−n) Ωn(n),OX(−n+ 1) Ωn−1(n− 1), . . . ,OX(−1) Ω1(1),OX×X = OX OX

Recalling that the Fourier-Mukai transform defined in 1.3.1 is functorial with respect to thekernel and preserves exact triangles, we obtain for any object A ∈ Db(X) a sequence of exacttriangles in Db(X):

ΦOX(−n)Ωn(n)(A)→ ΦOX(−n+1)Ωn−1(n−1)(A)→ ΦCn−1(A)→ ΦOX(−n)Ωn(n)(A)[1]

ΦCn−1(A)→ ΦOX(−n+2)Ωn−2(n−2)(A)→ ΦCn−2(A)→ ΦCn−1(A)[1]

. . .

ΦC1(A)→ ΦOXOX (A)→ ΦO∆(A)→ ΦC1(A)[1]

(2.4)

On the other hand, ΦO∆(A) ' A (1.3.2), so we obtain that any object A ∈ Db(X) is in thesubcategory generated by the collection

ΦOX(−n)Ωn(n)(A),ΦOX(−n+1)Ωn−1(n−1)(A), . . . ,ΦOX(−1)Ω1(1)(A),ΦOXOX (A).

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Lemma 2.2.1. For any i = 0, . . . , n there is a natural isomorphism:

ΦOX(−i)Ωi(i)(A) ' Ωi(i)⊗L RΓ(Pn,A⊗OX(−i))

Proof. Using the fact that OX(−i) is a locally free sheaf, we can write:

ΦOX(−i)Ωi(i)(A) = Rq∗(p∗(A)⊗L p∗OX(−i)⊗L q∗Ωi(i)) '' Rq∗(q∗Ωi(i)⊗L p∗(A⊗OX(−i)))

Now, we apply projection formula:

ΦOX(−i)Ωi(i)(A) ' Ωi(i)⊗L Rq∗p∗(A⊗OX(−i))

Finally, consider the following diagram:

X ×X X

X Spec k

p

q x

x

where x : X → Spec k is the canonical morphism of X. Since all the involved maps areproper and flat, we can invoke flat base change to get:

Rq∗ p∗ = x∗ Rx∗

Now, the functor x∗ coincides with the global section functor Γ; hence x∗ Rx∗ sends anysheaf F to the sheaf RΓ(F) regarded as a constant sheaf on X. Then, also noting that weare working with locally free sheaves (hence tensor product is exact), we can write:

ΦOX(−i)Ωi(i)(A) ' Ωi(i)⊗RΓ(A⊗OX(−i)).

Now, the object RΓ(A⊗OX(−i)) is a true complex of locally free sheaves with cohomologygroups:

RjΓ(A⊗OX(−i)) := Hj(RΓ(A⊗OX(−i)))

Since every complex of vector spaces splits, we have

RΓ(A⊗OX(−i)) 'n⊕j=0

RjΓ(A⊗OX(−i))[−j]

where the second term is a complex with zero differentials. Then:

ΦOX(−i)Ωi(i)(A) 'n⊕j=0

Ωi(i)dim RjΓ(A⊗OX(−i))[−j]. (2.5)

We conclude that Db(X) is generated as a triangulated category by the collection

E = Ωn(n), . . . ,Ω1(1),OX

where the elements are regarded as complexes concentrated in degree 0.

19

If we exchange the morphisms p and q in the definition of the Fourier-Mukai transform,we get a dual result. The natural isomorphism (2.5) corresponds to:

ΦOX(−i)Ωi(i)(A) 'n⊕j=0

OX(−i)dim RjΓ(A⊗Ωi(i))[−j]. (2.6)

Then, Db(X) is generated as a triangulated category by the collection

D = OX(−n), . . . ,OX(−1),OX.

The following lemma shows that the collection of sheaves D is actually a complete strongexceptional sequence for Db(X).

Lemma 2.2.2. For any pair of integers i, j in 0, . . . , n and for any integer m we have

HomDb(X)(OX(−i),OX(−j)[m]) = Hm(X,OX(i− j)) =

0 if m 6= 0

Symi−j(V ∗) if m = 0

where V is the k-vector space of dimension n+ 1 such that X = P(V ) and V ∗ is its k-dual.In particular, HomDb(X)(OX(−i),OX(−i)) ' k, so the objects OX(−i) are exceptional

and the sequence OX(−n), . . . ,OX(−1),OX is strong exceptional.

Proof. Recall that OX(−j)[m] is by definition equal to Tm(OX(−j)) where T is the transla-tion endofunctor of the category Db(X). Then, by [Wei94, Definition 10.7.1], the Hom groupHomDb(X)(OX(−i),OX(−j)[m]) coincides with the mth hyperext Extm(OX(−i),OX(−j)).Since the category of quasi coherent sheaves has enough injectives and the derived categoryof coherent sheaves is a full subcategory of the derived category of quasi coherent sheaves,we have ([Wei94, Theorem 10.7.4]):

Extm(OX(−i),OX(−j)) ' Hn(RHom(OX(−i),OX(−j)))

where RHom is the derived Hom. Since OX(−i) and OX(−j) are locally free, Hom is exactand then:

Extm(OX(−i),OX(−j)) ' Hm(RHom(OX(−i),OX(−j))) =

= Hm(Hom(OX(−i),OX(−j))) '' Hm(X,Hom(OX(−i),OX)⊗OX(−j)) '' Hm(X,OX(−i)∨ ⊗OX(−j)) '' Hm(X,OX(i)⊗OX(−j)) '' Hm(X,OX(i− j)).

The assertion hence follows from the classical result on cohomology groups of the twistedsheaf over the projective space.

A similar result holds also for the sheaves Ωi(i), showing that also the collection E iscomplete and strong exceptional in Db(X) (see [Bei78, Lemma 2]).

Lemma 2.2.3. For any pair of integers i, j in 0, . . . , n and for any integer m we have

HomDb(X)(Ωi(i),Ωj(j)[m]) =

0 if m 6= 0

∧i−j(V ∗) if m = 0

20

Then, we conclude with the following theorem:

Theorem 2.2.4. Consider the collection of sheaves over X

D = OX(−n), . . . ,OX(−1),OXE = Ωn(n), . . . ,Ω1(1),OX

Then, Db(X) is equivalent as a triangulated category to Kb(D) (resp. Kb(E)) i.e. the homo-topy category of finite complexes of sheaves whose objects consists of finite direct sums of theelements of D (respectively, of E).

Proof. The theorem follows easily from Proposition 1.2.9 and the fact that OX(−n), . . . ,OX(−1),OX(resp. Ωn(n), . . . ,Ω1(1),OX) is a complete strong exceptional sequence (hence a strong ex-ceptional set), as proved in the last lemma and the previous discussion.

21

Chapter 3

Derived category on the Grassmanian

The main point of Beilinson’s work was his idea of using the resolution of the diagonal tocompute a collection of generators for the derived categories on the projective space. In theyears that followed the publication of his paper, mathematicians extended this idea to thesituation of other homogenous schemes. One of the first examples is the work of Kapranovof 1985 ([Kap85]) where the case of Grassmanian varieties is considered.

In this chapter, following [Kap85], we compute the resolution of the diagonal as the Koszulcomplex associated to a section of a particular vector bundle. An introductory paragraphabout sheaf cohomology on Grassmannians and explicit computations for the zero schemeassociated to the section has been added.

3.1 Sheaf cohomology on the Grassmannian

Let k any algebraically closed field of characteristic 0 and let V a vector space of dimensionn over k. After the choice of a basis, we identify V with kn.

The Grassmannian G = G(d, V ) = G(d, n) of d-dimensional subspaces of V can bedefined in many different ways. A standard way (see [EH16, Section 3.2.1]) is to define theGrassmannian as the set of subspaces of V of dimension d and then consider an injective map,called Plucker embedding, from this set to the projective space P(∧dV ). The image of thePlucker embedding results to be a zero set of a particular ideal of homogeneous polynomials,whence the Grassmannian inherit the structure of projective smooth algebraic variety. Amore sophisticated approach (see [EH16, Section 3.2.2] or [EF05]) is to see the Grassmannianas a quotient of a space of matrices. Following [EF05], consider Ud,n ⊂ Adn the open setparametrizing matrices of dimension n × d with maximal rank d. The linear group GLd ofinvertible matrices of dimension d acts freely on Ud,n by multiplication on the right. Then,the Grassmannian can be defined as the quotient G(d, V ) = Ud,n/GLd. The space G(d, V )is a true Grasmsannian, since any element [M ] in the quotient defines uniquely a subspaceof V , i.e. the one spanned by the columns of the representative M . The structure of varietyover k is defined by glueing (see [Kol04, Chapter 1.4]).

Given any left GLd-module W , we can define a corresponding vector bundle on G :

W := [(M,w)] ∈ Ud,n ×W |[(M,w)] = [(Mg, gt · w)] for any g ∈ GLd

(transposition is necessary since W is a GLd-module on the left), and

π :W −→ G

[(M,w)] 7−→ [M ].

22

The canonical action of GLn on Ud,n by multiplication on the left makes W into a GLn-equivariant vector bundle with respect to the canonical action of GLn on G. This means thefollowing:

a) The map π :W → G is equivariant with respect to the action of GLn; indeed:

GLn ×W −→W(g, [(M,w)]) 7−→ [(gM,w)]

so π(g · [(M,w)]) = [gM ] = g · [M ] = g · π([(M,w)]).

b) for any point [M ] ∈ G, the action induced on the fibers π−1([M ]) → π−1(g · [M ]) is alinear map (actually, an isomorphism):

W([M ]) −→W([gM ])

[M,w] 7−→ [gM,w].

In particular, the cohomology groups of W are in their turn left GLn-modules. For example,consider the 0-th group of cohomology, i.e. global sections. We have a canonical identification:

H0(G,W) = f : Ud,n →W | f(Mg) = gt · f(M)

Then, for any h ∈ GLn, we set (h · f)(M) := f(htM). This gives a well-defined structure ofGLn-module on the left, since:

(g · (h · f))(M) = (h · f)(gtM) =

= f(htgtM) = f((gh)tM) = ((gh) · f)(M).

The description given so far allows us to make computations of sheaf cohomology onthe Grassmannian making use of the Borel–Weil–Bott theorem for flag varieties (see [FH04,§23.3] for a classical reference, [Lur] for a short proof).

Denote with Td ⊂ GLd the subgroup of diagonal matrices, Bd ⊂ GLd the subgroup ofupper triangular matrices and Ud ⊂ Bd the subgroup of upper triangular matrices with 1on the diagonal (so that Td ' Bd/Ud). The quotient Fld = GLd/Bd (with an opportunegeometric structure) is known as the complete flag variety of spaces of dimensions d. Itparametrizes “complete flags”, i.e. sequences of d + 1 strictly nested subspaces in a vectorspace of dimension d:

F0 = 0 ⊂ F1 ⊂ · · · ⊂ Fd ' kd

For any non-increasing sequence of integers λ = (λ1, . . . , λd) of length d (i.e. a Young diagramwith d rows) we consider the linear representation:

Td −→ End(k) ' k∗t1 00 t2

. . .td

7−→ tλ11 tλ2

2 · · · tλdd

23

which extends to a linear representation of Bd via the linear map Bd → Bd/Ud ' Td. Thisrepresentation induces on k a structure of left Bd-module. Then we can define on Fld theline bundle Lλ = GLd ×Bd k (where Bk acts diagonally, as above). We set:

V(d)λ := H0(GLd/Bd, Lλ)

which, as in the case of bundles over Grassmannian, has a structure of left GLd-module.Then, denote with Vλ the corresponding vector bundle on the Grassmannian.

The Borel–Weil–Bott theorem for sheaf cohomology on flag varieties can be specializedto the following theorem on Grassmannians:

Theorem 3.1.1. ([EF05, Theorem 2.2])Let λ = (λ1, . . . , λd) a sequence of non-increasing integers and let V (d)

λ and Vλ defined asabove. If λd ≥ 0, then:

a) H0(G,Vλ) = V(n)λ as left GLn-modules, where V (n)

λ is the GLn-module of highest weight(λ, 0) = (λ1, . . . , λd, 0, . . . , 0);

b) H i(G,Vλ) = 0 for all i > 0.

Remark 3.1.2. The GLn-module V (n)λ of highest weight (λ, 0) = (λ1, . . . , λd, 0, . . . , 0) can

also be computed applying to V the Schur functor associated to (λ, 0), namely Σ(λ,0).

We want to apply Theorem 3.1.1 to compute global sections of the hyperplane bundle ofthe Grassmannian. In the following discussion we fix the convention that, for any m > 0, theelements of the vector space km are column vectors.

First, we recall the trivial bundle V on G:

V := G× kn −→ G

([M ], w) 7−→ [M ].

The tautological bundle S on G is defined as the quotient:

S := [(M,w)] ∈ Ud,n × kd|[(M,w)] = [(Mg, g−1w)] for any g ∈ GLd

endowed with a well-defined injective map:

S → V[(M,v)] 7→ ([M ],Mv)

that identifies S as the subbundle of V whose fiber at [M ] coincides with the subspace of Vassociated to [M ] itself by definition of G.

The dual of the trivial bundle V∗ is defined as:

V := G× (kn)∗ −→ G

([M ], wt) 7−→ [M ].

(where wt denotes a transposed column vector, i.e. a row vector). The dual of the tautologicalbundle, also known as the hyperplane bundle, can hence be defined as the quotient:

S∗ := [(M,wt)] ∈ Ud,n × (kd)∗|[(M,wt)] = [(Mg,wtg)] for any g ∈ GLd

24

which is endowed with a well-defined, surjective map:

V∗ S∗

([M ], vt) 7→ [(M,vtM)]

that, on the fiber at [M ], corresponds to the restriction of a linear form on V to a linear formon the subspace associated to [M ].

The tautological quotient bundle V/S is defined as the cokernel of the injection S → V.Its fiber at any point [M ] is the quotient of V modulo the subspace associated to [M ].

Proposition 3.1.3. Let S∗ and V/S be respectively the hyperplane bundle and the tautologicalquotient bundle on G. Then there are natural isomorphisms:

H0(G,S∗) ' V ∗

H0(G,V/S) ' V

as left GLn-modules.

Proof. We apply Theorem 3.1.1 to the case λ = (1, 0, . . . , 0). The associated GLd-moduleV

(d)λ coincides, by the classical Borel-Weil-Bott theorem, with the defining left representation

of GLd, i.e.:

GLd × kd → kd

(g, v) 7→ gv.

The corresponding bundle Vλ on G is:

Vλ := [(M,w)] ∈ Ud,n × kd|[(M,w)] = [(Mg, gtw)] for any g ∈ GLd

endowed with the projection on G. A global section for Vλ is given by a map f : Ud,n →kd such that f(Mg) = gtf(M) for any matrix g ∈ GLd. Now, theorem 3.1.1 states thatH0(G,Vλ) is isomorphic to the GLn-module of highest weight λ = (1, 0 . . . , 0), hence Vwith the standard action of GLn by product on the left. We can describe the isomorphismV → H0(G,Vλ) as follows. If v ∈ kn is any column vector, the corresponding global sectionof Vλ is given by the mapping:

fv : Ud,n −→ kd

M 7−→M tv

that satisfies fv(Mg) = gtM tv = gtfv(M). For any matrix g ∈ GLn acting on kn, the globalsection corresponding to gv is:

fgv : Ud,n −→ kd

M 7−→M tgv

which coincides with the section g · fv given by the action of GLn on H0(G,Vλ),

g · fv : Ud,n −→ kd

M 7−→ fv(gtM) = (gtM)tv = M tgv

showing that the map v 7−→ fv is a true morphism of GLn-modules (actually, an isomor-phism).

25

We are now ready to prove the first part of the assertion. Notice that the bundle Vλ canbe identified canonically with the hyperplane bundle S∗ via the map:

Vλ −→ S∗

[(M,w)] 7−→ [(M,wt)].

On the other hand, there is an isomorphism of GLn-modules:

kn −→ (kn)∗

v 7−→ vt

where any matrix g ∈ GLn acts on (kn)∗ by g · vt := vtgt. The previous isomorphismV → H0(G,Vλ) hence translate into the following isomorphism of GLn-modules:

V ∗ −→ H0(G,S∗)vt 7−→ (fv : M 7→ vtM)

which shows the first part of the assertion.

The second part is proved with the aid of the following construction. If W ⊂ V is asubspace of dimension d, we have an exact sequence:

0→W → V → V/W → 0

and a dual exact sequence:

0→ (V/W )∗ → V ∗ →W ∗ → 0.

In particular, the above exact sequence shows that (V/W )∗ can be identified canonically withthe subset of V ∗ given by linear forms on V which are equal to zero when restricted to W .Then (V/W )∗ is a subspace of V ∗ of dimension n−d, i.e. defines a point of the GrassmannianG′ = G(n− d, V ∗). In other words, we have a mapping:

G = G(d, V ) −→ G(n− d, V ∗) = G′

W 7−→W⊥ = (V/W )∗

which results to be an isomorphism of smooth algebraic varieties. Under this isomorphism,the tautological quotient bundle V/S on G corresponds to the dual of the tautological bundleS ′ on G′. The theorem hence follows from the first part.

3.2 The resolution of the diagonal

We are now ready to compute a resolution of the diagonal as the Koszul complex associ-ated to a global section of a vector bundle. Let G × G be the product of two copies of theGrassmanian endowed with the projections p and q on the first and on the second componentrespectively. The diagonal embedding i : G→ G×G is a closed morphism which restricts toan isomorphism with the diagonal ∆ ⊂ G×G. Denote with OG the sheaf of regular functionsover G; we consider the diagonal sheaf to be the coherent sheaf O∆ = i∗OG×G.

Lemma 3.2.1. H0(G×G, p∗(S∗)⊗ q∗(V/S)) ' End(V )

26

Proof. Recall the Kunneth formula: for any pair of quasi-coherent sheaves F and L on G,and for any index i ≥ 0, there is a natural isomorphism:

H i(G×G, p∗F ⊗ q∗L) '⊕j+`=i

Hj(G,F)⊗k H`(G,L)

(For a proof of the formula, see [Kem93, Proposition 9.2.4] or [Gro63, 6.7.8]). Apply theformula for i = 0, F = S∗, L = V/S:

H0(G×G, p∗(S∗)⊗ q∗(V/S)) ' H0(G,S∗)⊗H0(G,V/S). (3.1)

By Proposition 3.1.3, we have hence an isomorphism:

H0(G×G, p∗(S∗)⊗ q∗(V/S)) ' V ∗ ⊗ V.

The assertion hence follows directly considering the canonical isomorphism V ∗⊗V ' End(V ).

The following key lemma gives us a regular section whose corresponding Koszul complexyields a resolution of the diagonal.

Lemma 3.2.2. Consider the global section ζ for the line bundle p∗(S∗) ⊗ q∗(V/S) corre-sponding to the identity map in EndV under the isomorphism of the previous lemma. Then,the zero scheme of ζ is the diagonal ∆ ⊂ G×G and ζ is a regular section.

Proof. Denote with e1, . . . , en the elements of the canonical basis for V = kn. Under theisomorphism V ∗ ' H0(G,S∗), denote with si the section of S corresponding to the elemente∗i of the dual basis of V . If [M ] ∈ G is an element of the Grassmannian, represented by arank-d matrix M of dimension n× d, the corresponding subspace W ⊂ V is spanned by thelinearly independent columns m1, . . . ,md of the matrix M . Then, the fibre of S∗ over [M ]is the dual space W ∗, which has a basis m∗1, . . . ,m∗d dual to m1 . . . ,md. The section si of S∗sends any point [M ] to the linear form which, under the dual basis m∗j , is represented by therow vector [0 · · · 1i · · · 0]M ; in other words: si([M ]) =

∑dj=1w

jim∗j where M = [wji ]i,j .

Conversely, under the isomorphism V ' H0(G,V/S), the element ei corresponds to asection ti whose image at any point [M ] ∈ G (where the columns of M span a subspaceW ⊂ V ) is the vector ei regarded as an element of the quotient vector space V/W .

The identity morphism idV ∈ End(V ) corresponds to the element∑n

i=1 e∗i ⊗ ei ∈ V ∗⊗ V

which in its turn corresponds to the element∑

i si ⊗ ti ∈ H0(G,S∗) ⊗H0(G,V/S). UnderKunneth isomorphism (3.1), this defines a global section ζ of p∗(S∗) ⊗ q∗(V/S). For anypoint ([M ], [L]) ∈ G×G, where [M ] and [L] represents respectively the subpaces W,U ⊂ Vof dimension d, the image of ζ at ([M ], [L]) is:

ζ([M ], [L]) =∑i

si([M ])⊗ ti([L]) ⊂W ∗ ⊗ V/U.

If the point ([M ], [L]) belongs to the diagonal ∆ ⊂ G×G, then [M ] = [L]. Without lossof generality we can choose the representatives to be equal, i.e. M = L is single n× d matrix

27

whose columns span W = U ⊂ V . Then:

ζ([M ], [M ]) =∑i

si([M ])⊗ ti([M ]) =

=∑i

∑j

wijm∗j

⊗ ei =

=∑j

m∗j ⊗

(∑i

wij ei

)=

=∑j

m∗j ⊗ mj = 0 ∈W ∗ ⊗ V/W

since mj ∈ W . If we denote with Z(ζ) the zero scheme of ζ and with |Z(ζ)| the underlyingtopological subspace, this computation shows that |∆| ⊂ |Z(ζ)|.

Finally, we have prove that Z(ζ) = ∆ as closed subschemes of G ×G; by the character-ization of closed subschemes, is enough to check that the ideals of sheaves corresponding toZ(ζ) and ∆, denoted respectively with Z ⊂ OG×G and J ⊂ OG×G, are equal. We show thisequality locally on the affine charts of the product scheme G×G. For any fixed tuple indexesI = (i1, . . . , in), 1 ≤ i1 < · · · < in ≤ n, consider the subset of Un×d consisting of those matrixM whose I-th submatrix MI , obtained taking only the rows (i1, . . . , in), has rank d. Thesubsets UI , for I varying among all possible multi-indexes, cover the whole set Un×d. Bydefinition, the quotients GI = UI/GLd form an open cover of the Grassmannian scheme G(see [Kol04, Chapter 1.4]). In particular, they define an affine open cover for G. For example,consider the case I = (1, . . . , d). Any element [M ] ∈ GI has a unique representative of theform

M(MI)−1 =

1 0 . . . 00 1 . . . 0...

......

0 0 . . . 1m11 m12 . . . m1d...

......

mn−d,1 mn−d,2 . . . mn−d,d

=

[Id×dM ′

]

where M ′ = [mij ] is any (n− d)× d matrix. The open subset GI ⊂ G then results an affinechart thanks to the identification of the space of matrices of dimension (n− d)× d with theaffine scheme Ad(n−d) = Spec(k[. . . xij . . . ]), where the variables xij correspond to the entriesmij of the matrix M ′.

The product scheme G × G has a cover given by the affine open subsets GI × GJ forI and J varying among all possible multi-indexes. For simplicity, we do our computationin the case I = J = (1, . . . , d). We consider two copies of GI , isomorphic respectively toSpec(k[. . . , xij , . . . ]) and Spec(k[. . . , yij , . . . ]). The evaluation at GI ×GI of the sheaf idealof the diagonal is equal to:

J (GI ×GI) = (xij − yij)i,j ⊂ k[. . . , xij , . . . , yij , . . . ] = OG×G(GI ×GI)

On the other hand, the sheaf ideal corresponding to the zero scheme of ζ is computed bydefinition as the image of the following evaluation map:

ζ∨ : p∗(S)⊗ q∗((V/S)∗) −→ OG×G

28

Let’s look at how the bundles S and (V/S)∗ appear when restricted to GI ' Ad(n−d). Thefiber of S over the point (mij) is the subspace of V generated by the columns of the associatedmatrix M , i.e. the subspace generated by the vectors sj = ej +

∑n−di=0 mijej+d for j going

from 1 to d. The fiber of (V/S)∗ over the same point (mij) is given by those linear forms onV that are zero on the subspace generated by the columns of the associated matrix M . Abasis for this fiber is given by the linear forms `i = e∗i+d −

∑dj=0mije

∗j for i going from 1 to

(n− d). Indeed, the evaluation of `i at sj is equal to:

`i(sj) =

e∗i+d − d∑j=0

mije∗j

(ej +n−d∑i=0

mijej+d

)= mij −mij = 0

showing that `i ∈ (V/S)∗((mij)); as different terms e∗i+d appear for i varying from 1 to (n−d),the `i are (n − d) linearly independent vectors in a space of dimension (n − d), hence theyform a basis.

Working in the local coordinates of GI × GI , we conclude that the bundle p∗(S) ⊗q∗((V/S)∗) can be seen locally as a subbundle of the trivial bundle p∗(V) ⊗ q∗(V∗) witha basis given by the elements:

sj ⊗ `i =

(ej +

n−d∑i=0

xijej+d

)⊗

e∗i+d − d∑j=0

yije∗j

.

The map ζ∨ : p∗(S)⊗ q∗((V/S)∗) −→ OG×G, coming from id ∈ End(V ), corresponds locallyto the evaluation map. Then, its image is the ideal generated by:

`i(sj) =

e∗i+d − d∑j=0

yije∗j

(ej +n−d∑i=0

xijej+d

)= xij − yij

which are the generators for J (GI ×GI). We conclude that Z = J locally on affine charts,and hence globally. The fact that ζ is a regular section follows from dimension count as inthe case of Pnk .

Let S⊥ := (V/S)∗. Taking the Koszul complex associated to the section ζ, we get aresolution for the diagonal sheaf O∆:

· · · → Λ2(p∗S ⊗ q∗(S⊥))→ Λ2(p∗(S∗)⊗ q∗(S⊥))→ OG×G → O∆ (3.2)

By Lemma 1.5.2,

Λi(p∗(S∗)⊗ q∗(S⊥)) '⊕|α|=i

Σα(p∗S)⊗ Σα∗(q∗S⊥) =

=⊕|α|=i

p∗(ΣαS)⊗ q∗(Σα∗S⊥)

Remark 3.2.3. If W is a k-vector of dimension n and the Young diagram α has more thann rows, then ΣαW is zero ([FH04, Theorem 6.3.1]). Then, the direct sum

⊕|α|=i p

∗(ΣαS)⊗q∗(Σα∗S⊥) can be restricted only to those terms for which the diagram α that have a numberof rows less or equal than dimS = k, and a number of columns less or equal than dim(S⊥) =n− k. In particular, it follows that the direct sum is empty (i.e. is equal to the zero sheaf)when i > n(n− k). Then, the resolution of the diagonal (3.2) is bounded.

29

We conclude that the sheaf O∆ admits a bounded resolution:

· · · →⊕|α|=2

p∗(ΣαS)⊗ q∗(Σα∗S⊥)→⊕|α|=1

p∗(ΣαS)⊗ q∗(Σα∗S⊥)→ OG×G → O∆ (3.3)

3.3 The derived category Db(G)

Similarly to the case of Pnk , the resolution of the diagonal (3.3) allows us to find twofamilies of generators for Db(G). Let A any object of Db(G), and consider the usual Furier-Mukai with generic kernel E :

ΦEX→X : Db(X) −→ Db(X)

ΦEX→X(−) = Rp∗(q∗(−)⊗L E).

Using the exact triangles arising from the resolution of the diagonal, we obtain that A 'ΦO∆(A) is in the triangulated category generated by the elements of the form:

Φ⊕|α|=i p

∗(ΣαS)⊗q∗(Σα∗S⊥)(A).

Then we have a sequence of natural isomorphisms:

Φ⊕|α|=i p

∗(ΣαS)⊗q∗(Σα∗S⊥)(A) =

= Rp∗(q∗(A)⊗L

(⊕|α|=ip∗(ΣαS)⊗ q∗(Σα∗S⊥)

))=

= Rp∗(q∗(A)⊗

(⊕|α|=ip∗(ΣαS)⊗ q∗(Σα∗S⊥)

))=

= Rp∗(⊕|α|=ip∗(ΣαS)⊗ q∗(A⊗ Σα∗S⊥)

)=

=⊕|α|=i

Rp∗(p∗(ΣαS)⊗ q∗(A⊗ Σα∗S⊥)

)'

'⊕|α|=i

ΣαS ⊗RΓ(A⊗ Σα∗S⊥)

)'

'⊕j

⊕|α|=i

(ΣαS)dim RjΓ

(A⊗Σα

∗S⊥))

[−j]

We conclude that Db(G) is generated as a triangulated category by the sheaves of the formΣαS where α is such that Σα∗S⊥ 6= 0.

Actually, the set of sheaves ΣαS | Σα∗S⊥ 6= 0 generating Db(G) is a strong exceptionalset. This gives a more detailed description of Db(G) in terms of the homotopy categoryKb(ΣαS | Σα∗S⊥ 6= 0).

Lemma 3.3.1. The set of sheaves D := ΣαS | Σα∗S⊥ 6= 0 is a strong exceptional set.

Proof. We give only a sketch of the proof, referring to [Kap85, §2] for more details.First, we introduce a slight generalization of Schur functors. If α = (α1, . . . , αm) is any

non-increasing sequence of integers (also with negative entries) andW is a k-vector space, wecan define functorially ΣαW as the irreducible representation of GL(W ) with highest weightα. If we set

−α = (−αm, . . . ,−α1)

30

then

(ΣαW )∗ ' Σα(W ∗) ' Σ−α(W ).

To prove the lemma, proceed as follows. If ΣαS and ΣβS are in the set of generators D,then

HomDb(G)(ΣαS,ΣβS[m]) = Extm(ΣαS,ΣβS) '

' Hm(RHom(ΣαS,ΣβS)) =

= Hm(Hom·(ΣαS,ΣβS)) '' Hm(G,Hom(ΣαS,OG)⊗ ΣβS) =

= Hm(G,Σ−αS ⊗ ΣβS)

Since α and β have less than n − k rows, from the rule for multiplying representations weobtain that every summand ΣγS appearing in the decomposition of Σ−αS⊗ΣβS is such thatany element γi of the sequence lies between −(n− k) and n− k. The assertion hence followsfrom the following Proposition ([Kap85, Proposition 2.2, b]):

Proposition 3.3.2. Let γ = (γ1, . . . , γm) a sequence of integers such that γ1 ≥ . . . γm ≥−(n − k). Then Hm(G,ΣγS) = 0 for any m 6= 0 and H0(G,ΣγS) 6= 0 only when all γi areless or equal than 0. In this case,

H0(G,ΣγS) = Σ−γV ∗

Theorem 3.3.3. ([Kap85, Proposition 1.4]) The derived category of the GrassmannianDb(G) is equivalent as a triangulated category to Kb(D) i.e. the homotopy category of fi-nite complexes of sheaves whose terms are finite direct sums of the sheaves ΣαS such thatΣα∗S⊥ 6= 0.

Dually, let E := ΣαS⊥ | Σα∗S 6= 0. Then, the derived category of the GrassmannianDb(G) is equivalent as a triangulated category to Kb(E) i.e. the homotopy category of finitecomplexes of sheaves whose terms are finite direct sums of the sheaves ΣαS⊥ such that Σα∗S 6=0.

Proof. The first part of the statement is a consequence of the discussion above together withProposition 1.2.9. The second part of the lemma is obtained considering the isomorphismbetween G(k, V ) and G(n − k, V ∗), under which the tautological bundle S corresponds toS⊥.

31

Chapter 4

Derived category on the Quadric

Following the work of Beilinson about the projective space ([Bei78]) and the work ofKapranov about the Grassman varieties ([Kap85]), another paper of Kapranov was pub-lished in the 1986 announcing the extension of Beilinson’s technique to the case of Quadrics([Kap86]). All these results were then collected by Kapranov in a more general paper, Onthe derived category of coherent sheaves on some homogeneous spaces ([Kap88]), where thecases of flag varietes and incidence varieties were also considered. In this chapter, we discussthe case of quadrics. We will follow mainly [Kap88], adding clarifications and computationswhere they were not explicited in the original paper.

4.1 Quadrics and Clifford algebras

Let n a positive integer, N = n+ 1, and let E a N -dimensional vector space over a fixedfield k of characteristic 0. Consider the projective space over k, X = P(E) = Pnk endowedwith the usual sheaf of regular function OX .

Denote with (E ⊗ E)∗ the k-dual vector space of E ⊗ E and let f ∈ (E ⊗ E)∗ a non-degenerate symmetric bilinear form. It induces a quadratic form

q : E → k

q(x) = f(x⊗ x)

that can be thought as an homogeneous polynomial of degree 2 in N variables.

We define the (non-degenerate) quadric associated to f (or to q) to be the closed sub-scheme ofX obtained as the zero set of the homogeneous polynomial q. The graded coordinatealgebra of Q can be recovered as B =

⊕H0(Q,OX(i)).

Remark 4.1.1. B is defined only in non-negative degrees. Indeed if x0, . . . , xn is a basis forE, and if we denote with ξ0, . . . , ξn the associated dual basis for E∗, then clearly B is equal,as a graded algebra, to the quotient of the symmetric algebra k[ξ0, . . . , ξn] by the principalhomogeneous ideal I = (q).

Associated to Q there is also a non-commutative graded algebra which plays an importantrole in the construction of the resolution for the diagonal in the case of quadrics.

Definition 4.1.2. The graded Clifford algebra associated to the bilinear form f is thegraded k-algebra A = A(E, f) generated by the elements of E, considered as generators in

32

degree 1, and by a special element h of degree 2, together with relations

uv + vu = 2f(u, v)h for any pair of vectors u, v ∈ Evh = hv for any vector v ∈ E.

Remark 4.1.3. The algebra A is graded and the homogeneous components with negativedegrees are zero, so A = ⊕i≥0Ai. By definition, A1 coincides with the vector space E.

4.2 The Generalized Koszul resolution

Maintaining the notations of the previous paragraph, we introduce now a resolution forthe base field k, regarded as a B-module. First of all, denote with B+ the ideal in B generatedby the elements in positive degrees (B+ = ⊕i>0Bi). Then, the field k can be regarded as aB module via the identification k ' B/B+.

The resolution involves the following elements.For any index i ≥ 0 denote by A∗i the k-dual of the homogeneous component of degree i

in the graded algebra k.Fix then a basis x0, . . . , xn for the vector space E, orthogonal with respect to the non-

degenerate symmetrical bilinear form f , and denote with ξ0, . . . , ξn the associated dualbasis for E∗.

Finally, for any j = 0, . . . , n, consider the map `ξj : B → B of multiplication by ξj onthe left and the map r∗xj : A∗i+1 → A∗i defined as the dual of the map rxj : Ai → Ai+1 ofmultiplication by xj on the right.

Proposition 4.2.1. There exists an exact resolution for the B-module k

· · · → A∗2 ⊗B → A∗1 ⊗B → A∗0 ⊗B → k (4.1)

where the differential at any position is equal to∑n

j=0 r∗xj ⊗ `ξj .

The proof of the proposition involves a construction in homological algebra, called Koszul-Tate resolution, which is a generalization of the Koszul resolution introduced in Chapter 1.This construction was introduced by Tate in his paper [Tat57]. The application of thisconstruction to the case of quadrics was presented by Richard Swan in his paper aboutK-Theory of Quadric Hypersurfaces ([Swa85, § 7]

To cite the result of Tate, we first need some notation. Four our purpose, consider R acommutative, noetherian k-algebra.

Definition 4.2.2. A differential graded algebra (DG-algebra) C· is a family of R-modules Cii≥0 endowed with:

• a bilinear product Ci ⊗Cj → Ci+k and an element 1 ∈ C0 such that ⊕Ci is a (graded)associative R-algebra with unit;

• a family of R-differentials d : Ci+1 → Ci, i.e. linear maps of R-modules satisfying thecondition d2 = 0 and the following Leibniz rule, for any a ∈ Ci and b ∈ Cj :

d(a · b) = d(a) · b+ (−)ia · d(b).

When there is no confusion, we will refer to a DG-algebra just by mentioning its associatedgraded algebra C = ⊕i≥0Ci

We call a DG-algebra C skew-commutative if ab = (−)deg adeg bba holds for any a, b ∈ C.

33

Any DG-algebra C yields an obvious cochain complex, with objects the Ci and with chainmaps the differentials d : Ci+1 → Ci. We call C acyclic if the associated cochain complex isexact.

Remark 4.2.3. The Koszul-resolution described (1.7) is just the cochain complex associatedto the exterior algebra of the free module E over the ring A, seen as a DG-algebra.

We will now state a theorem, which is a particular case of [Tat57, Theorem 4].

Theorem 4.2.4. Let R a commutative, noetherian k-algebra. Let t0, . . . , tn a regular sequenceof elements in R and let M = (t0, . . . , tn) the two-sided ideal generated by those elements.Suppose that q ∈ M is a non-zero divisor in R and write q =

∑ni=0 citi. Let J = (q) and

denote with R, M respectively the quotients R/J and M/J .Consider the differential graded algebra C = R〈T0, . . . , Tn, Q〉 with generators Ti in degree

1 and a generator Q in degree 2 and with d(Ti) = ti, d(Q) =∑n

j=0 cjTj. Denoting with [Tj ]and [Qs] the free k-modules generated by Tj and Qs respectively, the component of degree i inC is equal to

Ci =⊕

r+2s=i

Λr

n⊕j=0

[Tj ]

⊗k [Qs]

⊗k Rand the differential is extended over all C with the aid of the Leibniz rule.

Then the DG-algebra C is acyclic and the associated cochain complex is a free left reso-lution of R/M as an R-module.

Under the notations of Proposition 4.2.1, we apply Tate’s theorem to R = k[ξ0, . . . , ξn],M = (ξ0, . . . , ξn) and q equal to the quadratic form regarded as an homogeneous polynomialof degree two in the ξi, so that the ideal J in the statement becomes the ideal I previouslydefined. Note that the basis x0, . . . , xn is orthogonal with respect to bilinear form, so wecan write q =

∑nj=0〈xj , xj〉ξ2

j .We have R = B and R/M ' R/M = B/B+ ' k. Then, the component of degree i

becomes:

Ci =⊕

r+2s=i

Λr

n⊕j=0

[Tj ]

⊗k [Qs]

⊗k B.The differential is defined by d(Ti⊗1⊗1) = 1⊗1⊗ξi and d(1⊗Q⊗1) =

∑nj=0 Tj⊗1⊗〈xj , xj〉ξj .

By Tate’s theorem, the cochain complex C·:

· · · → Ci+1 −→dCi → · · · → C0 → 0

is exact. We will use this information to prove the exactness of the resolution 4.1. To dothis, we need to consider a slight different version of the complex C·.Definition 4.2.5. We define a cochain complex C ′·, such that C ′i = Ci for any i ∈ Z and

d′i : C ′i → C ′i−1d′i = (−)i−1di

Claim. There exists an isomorphism of cochain complexes between C ′· and the complex (4.1):

· · · → A∗2 ⊗B → A∗1 ⊗B → A∗0 ⊗B → k

Proof. (of Claim). We start by defining a collection of isomorphisms A∗i ⊗B → C ′i, and thenwe show that these isomorphisms fits into an isomorphism of chain complexes.

34

Lemma 4.2.6. For any i ≥ 0, there is an isomorphism of k modules:

A∗i '⊕

r+2s=i

(Λr

(n⊕i=0

[Ti]

)⊗k [Qs]

)

Proof. (sketch). Recall that the basis x0, . . . , xn of E is orthogonal with respect to bilinearform f . Since the xi’s generate E as a k-vector space, then the k-algebra A is generated bythe xi’s (regarded as generators in degree 1), together with the central element h of degree 2.

From this observation, it follows that Ai is spanned as a k-module by the elements of theform xj1 . . . xji−2ph

p where 0 ≥ p ≥ i/2 and j1, . . . , ji−2p are arbitrary indexes in 0, . . . , n.Since by orthogonality xixj = −xjxi any time that i 6= j and xixi = 〈xi, xi〉h, we concludethat the indexes j1, . . . , ji−2p can be chosen such that 0 ≤ j1 < j2 < · · · < ji−2p ≤ n.

Since there are no other relations between the generators of the k-algebra A, it can beproved that for any i the set of elements xj1 . . . xjrhs | r + 2s = i and 0 ≤ j1 < j2 < · · · <jr ≤ n is a basis for Ai regarded as a k-module. The dual basis (xj1 . . . xjrhs)∗ is a basisfor A∗i .

Then, we can define a k-linear map out from A∗i just assigning the images of the elementsof the basis:

φi : A∗i −→⊕

r+2s=i

(Λr

(n⊕i=0

[Ti]

)⊗k [Qs]

)(xj1 . . . xjrh

s)∗ 7−→ (Tj1 ∧ · · · ∧ Tjr)⊗ (−)sQs/s!

The set of these images is in its turn a basis for the codomain, so the given map is anisomorphism.

Lemma 4.2.7. The collection of isomorphisms φ· = φii≥0 defines an isomorphism betweenthe complex (4.1) and the complex C ′·.

Proof. For any i ≥ 0, we have to prove that the following square commute:

A∗i+1 ⊗B A∗i ⊗B

Ci+1 Ci

φi+1⊗1

d1

φi⊗1

(−)id2

where the differential in the top side of the diagram is∑n

i=0 r∗xi ⊗ `ξi and the differential

d2 in the bottom side is the one induced by d2(Ti ⊗ 1⊗ 1) = 1⊗ 1⊗ ξi and d2(1⊗Q⊗ 1) =∑nj=0 Tj ⊗ 1⊗ 〈xj , xj〉ξj .Since all the maps appearing in the diagram are B-linear and k-linear, it is enough to check

that the square commutes for the elements of the form (xj1 . . . xjrhs)∗⊗1 where r+2s = i+1.

We show the computation for the element y = (x0 . . . xrhs)∗⊗1 ∈ A∗i+1⊗B (where r+2s = i);

the other cases are done similarly.The image of y through φi+1⊗1 is T0∧· · ·∧Tr⊗ (−)sQs/s!⊗1. Forgetting for a moment

35

the change of sign depending on i, we compute the image of (φi+1 ⊗ 1)(y) throught d2:

(d2 (φi+1 ⊗ 1))(y) = d2((1⊗ (−)sQs/s!⊗ 1) · (T0 ∧ · · · ∧ Tr ⊗ 1⊗ 1)) =

= (1⊗ (−)sQs/s!⊗ 1) · d2(T0 ∧ · · · ∧ Tr ⊗ 1⊗ 1)+

+ d2(1⊗ (−)sQs/s!⊗ 1) · (T0 ∧ · · · ∧ Tr ⊗ 1⊗ 1) =

=r∑l=0

(−)l(T0 ∧ · · · ∧ Tl ∧ · · · ∧ Tr ⊗ (−)sQs/s!⊗ ξl

)+

+ (−)r+1n∑

l=r+1

(T0 ∧ · · · ∧ Tr ∧ Tl ⊗ (−)sQs−1/(s− 1)!⊗ 〈xl, xl〉ξl

)Lifting (d2 (φi+1 ⊗ 1))(y) in A∗i ⊗B via (φi ⊗ 1)−1, we obtain:

((φi ⊗ 1)−1 d2 (φi+1 ⊗ 1))(y) =r∑l=0

(−)l ((x0 . . . xl . . . xrhs)∗ ⊗ ξl) +

+ (−)rn∑

l=r+1

((x0x1 . . . xr−1xrxlh

s−1)∗ ⊗ 〈xl, xl〉ξl)

Then, recalling that the differential appearing in the bottom side of the diagram is actually(−)id2 (equal to (−)rd2 since r + 2s = i), we have to consider:

((φi ⊗ 1)−1 (−)id2 (φi+1 ⊗ 1))(y) =r∑l=0

(−)r+l ((x0 . . . xl . . . xrhs)∗ ⊗ ξl) +

+ (−)2rn∑

l=r+1

((x0x1 . . . xr−1xrxlh

s−1)∗ ⊗ 〈xl, xl〉ξl)

=

=r∑l=0

(−)r+l ((x0 . . . xl . . . xrhs)∗ ⊗ ξl) +

+n∑

l=r+1

((x0x1 . . . xr−1xrxlh

s−1)∗ ⊗ 〈xl, xl〉ξl)

On the other side, we can compute:

d1(y) =n∑k=0

r∗xk((x0 . . . xrhs)∗)⊗ ξk

Direct computation shows that:

• if 0 ≤ l ≤ r, then the evaluation of r∗xl((x0 . . . xrhs)∗) at x0 . . . xl . . . xrh

s is equal to

(x0 . . . xrhs)∗(x0 . . . xl . . . xrh

s · xl) =

= (x0 . . . xrhs)∗((−)r−lx0 . . . xrh

s) = (−)r−l

and is zero at the other elements of the basis, so r∗xk((x0 . . . xrhs)∗ = (−)r−l(x0 . . . xl . . . xrh

s)∗;

• if l > r, then the evaluation of r∗xk((x0 . . . xrhs)∗) at x0 . . . xrxlh

s−1 is equal to

(x0 . . . xrhs)∗(x0 . . . xrxlh

s−1 · xl) =

= (x0 . . . xrhs)∗(〈xl, xl〉x0 . . . xrh

s) = 〈xl, xl〉

since xl·xl = 〈xl, xl〉h, and is zero on the other elements of the basis, so r∗xk((x0 . . . xrhs)∗ =

(x0 . . . xrxlhs−1)∗.

36

Then

d1(y) =

r∑l=0

(−)r−l ((x0 . . . xl . . . xrhs)∗ ⊗ ξl) +

+

n∑l=r+1

((x0x1 . . . xr−1xrxlh

s−1)∗ ⊗ 〈xl, xl〉ξl)

Since the parity of r − l and r + l is the same (the difference being 2l) we obtain:

d1 = (φi ⊗ 1)−1 (−)id2 (φi+1 ⊗ 1)

and so we have the assertion.

The proof of the claim ends the proof of the proposition.

Remark 4.2.8. Since B is a graded ring, we can take into consideration graded modulesover B. If k and A∗i are regarded as graded modules concentrated in degree 0, then we canslightly modify the resolution (4.1) to obtain a resolution of graded modules:

· · · → A∗2 ⊗B(−2)→ A∗1 ⊗B(−1)→ A∗0 ⊗B → k (4.2)

After this arrangement, each differential map becomes a homogeneous map of graded rings,in the sense that preserves the grading. Indeed, if b ∈ B has degree l, then it has degree l+ iin B(−i); for any a ∈ A∗i , the element a⊗ b ∈ A∗i ⊗B(−i) has degree l+ i. Its image throughthe differential is the element y =

∑r∗xj (a)⊗ ξjb. Since ξjb has degree l + 1 in B, it follows

that y has degree j + 1 + i− 1 = l + i in A⊗B(−i+ 1).

Remark 4.2.9. Since any A∗i is a k-module of finite rank, it follows that A∗i ⊗B(−i) is a freeB(−i)-module of finite rank. Then we have found a linear free resolution for k, regarded asa B-module. This implies that B is Koszul algebra, and the resolution 4.2 is the generalizedKoszul complex associated to B. Moreover, the algebras A and B are in Priddy duality, inthe sense that A ' ExtB(k, k) and B ' ExtA(k, k) as graded algebras. More details on thetheory of Koszul algebras can be found on the book by Polishchuk and Positselski ([PP05]).

4.3 The resolution of the diagonal

We come now to the core of this chapter. Similarly to the cases of projective space andGrassmanians, we want to find a bounded resolution for the diagonal of Q×Q. Unfortunately,in the case of quadrics the construction of such resolution is not as straightforward as in theprevious cases. The first step will be the definition of a resolution for the diagonal which willbe infinite on the right; this will be done in this section.

Consider first the following (non-exact) truncation of the complex (4.2), twisted by i:

A∗i ⊗B → A∗i−1 ⊗B(1)→ · · · → A∗2 ⊗B(i− 2)→ A∗1 ⊗B(i− 1)→ A∗0 ⊗B(i) (4.3)

This is an exact sequence of graded B-modules. Then, using the localization functor ofSerre ([Har77, Chapter II.5]) this defines an exact sequence of sheaves on Q:

A∗i → A∗i−1(1)→ · · · → A∗0(i) (4.4)

where A∗i−j(j) denotes the OQ-module corresponding to A∗i−j ⊗B(j).

37

Definition 4.3.1. Define Ψi to be the OQ-module equal to the kernel of A∗i → A∗i−1(1) inthe exact sequence (4.4).

Remark 4.3.2. The sheaves Ψi are locally free OQ-modules. For i = 0, Ψ0 = OQ so theassertion is trivial. For i = 1, note that k = 0, hence the map of sheaves A∗1(i− 1) → A∗0(i)coming from the exact sequence (4.2.1) is a surjective map of locally free OQ-modules; thenΨ1 is locally free. Splitting the exact sequence (4.4) into short exact sequences and arguingby induction, it follows that Ψi is locally free for all i.

Now let Q×Q be the product scheme of Q by itself endowed with the projections p, q onthe first and on the second component respectively.

Consider on Q × Q the sheaf Ci : p∗(Ψi) ⊗ q∗OQ(−i). For any element xj ∈ E = A1 ofthe basis, and for its corresponding element ξj ∈ E∗ = B1 in the dual basis, we consider themaps of multiplication on the left:

`xj : Ai → Ai+1

`ξj : B(−i− 1)→ B(−i)

The map `ξj induces by the localization functor of Serre a map of OQ-modules:

˜ξj : OQ(−i− 1)→ OQ(−i).

The dual of any map `xj is a map

`∗xj : A∗i+1 → A∗i

that can be extended to a map of B-modules `∗xj ⊗1 : A∗i+1⊗B → A∗i ⊗B and hence inducesa map of OQ-modules:

˜∗xj : A∗i+1 → A∗i .

We define analogously for any index l the maps:

˜∗xj : A∗i−l+1(l)→ A∗i−l(l).

Lemma 4.3.3. The collection of maps `∗xj defines map of chain complexes of OQ-modules:

A∗i+1 A∗i (1) . . .

A∗i A∗i−1(1) . . .

d

˜∗xj

d

˜∗xj

d d

Proof. Recall that the differential d is induced by the map of B-modules∑n

j=0 r∗xj ⊗ `ξj ,

while the vertical maps are induced by the map of B-modules `∗xj ⊗ 1. Now, in the algebra Athe operation of multiplication on the left commutes with multiplication on the right, hence: n∑

j=0

r∗xj ⊗ `ξj

(`∗xj ⊗ 1) = (`∗xj ⊗ 1)

n∑j=0

r∗xj ⊗ `ξj

Therefore the corresponding squares of maps of OQ-modules are commutative, so we get theassertion.

Corollary 4.3.4. The maps ˜xj can be restricted to maps ˜

xj : Ψi+1 → Ψi

38

Definition 4.3.5. We define C· to be the complex of sheaves on Q × Q whose objects arethe sheaves Ci = p∗(Ψi)⊗ q∗OQ(−i), for i ≥ 1, C0 = p∗(OQ)⊗ q∗(OQ) = OQ×Q and whosedifferentials are the maps:

dC : Ci+1 → Ci

dC =

n∑j=0

p∗(˜∗xj )⊗ q

∗(`ξj ).

The complex C· just defined provides an infinite left resolution of the diagonal, as statedin the following proposition:

Proposition 4.3.6. [Kap88, Proposition 4.6] Let i : Q→ Q×Q the diagonal embedding andlet ∆ ⊂ Q×Q the isomorphic image of Q through i (i.e. the diagonal of Q×Q). Then thecomplex of coherent sheaves C· on Q×Q

· · · → p∗(Ψ2)⊗ q∗OQ(−2)→ p∗(Ψ1)⊗ q∗OQ(−1)→ OY×Y

is a left infinite resolution of the coherent sheaf O∆ = i∗(OQ).

Proof. The proof is made with the aid of a double complex defined as follows.

First, note that (as a consequence of Segre embedding) the structure sheaf on Q × Q isobtained by Serre localization of the graded ring B2 := ⊕j≥0Bj ⊗ Bj ; and in general anysheaves of B2-modules yields a sheaf on Q ⊗ Q. Under this correspondence, for example,the sheaf p∗A∗i (l)⊗ q∗(B(−m)) can be obtained as the localization of the graded B2-module⊕j≥0(A∗i ⊗Bj+l)⊗Bj−m, where the structure of B2-module is suggested by the presence ofthe parenthesis.

We can therefore form a double complex D·· concentrated in the second quadrant:. . .

. . . 0

. . . ⊕j(A∗0 ⊗Bj+2)⊗Bj−2 0

. . . ⊕j(A∗1 ⊗Bj+1)⊗Bj−2 ⊕j(A∗0 ⊗Bj+1)⊗Bj−1 0

. . . ⊕j(A∗2 ⊗Bj)⊗Bj−2 ⊕j(A∗1 ⊗Bj)⊗Bj−1 ⊕j(A∗0 ⊗Bj)⊗Bj

Under the usual notation for the basis of E and E∗, the vertical maps are∑

i(r∗xi⊗`ξi)⊗1

and the horizontal maps are∑

i(`∗xi ⊗ 1)⊗ `ξi .

Consider first the columns ofD··. If we denote withD··j the double complex correspondingto the summand of degree j at any position, then the ith column of any D··j can be resumedas the graded part of the truncated generalized Koszul complex 4.3, tensored by the freeB-module B(−i). If follows that this column is exact and hence the complex of modules D··has exact columns.

Since Serre localization to is exact, the corresponding double complex D·· of sheaves onQ×Q has exact columns and indeed there is a canonical augmentation map C· → D··.

39

A standard argument in homological algebra (involving the cone of the augmentationmap, see for example [Wei94, Proof of Theorem 2.7.2]) shows that the complex of sheaves C·is quasi-isomorphic to the total complex of sheaves D· := Tot(D··).

On the other side, consider the B2-module B∆ = ⊕jB2j , where the structure of B-moduleis induced by the multiplication map Bj⊗Bj → B2j in the graded ring B. The corresponding(by Serre localization) sheaf B∆ on Q×Q can be naturally identified with O∆. We show thatthere is a canonical quasi-isomorphism of complexes φ from D· to B∆ (regarded as a complexconcentrated in degree 0). It follows that we have a commutative diagram of complexes ofsheaves on Q×Q:

C· O∆

D· B∆

and we conclude that there is a quasi-isomorphism C· → O∆, meaning that C· is a leftresolution for O∆.

To construct φ, consider again the double complex D··j for any j. The ith row startingfrom the bottom:

(A∗j−i ⊗Bj+i)⊗B0 → · · · → (A∗1 ⊗Bj+i)⊗Bj−i−1 → (A∗0 ⊗Bj+i)⊗Bj−i → 0

can be seen, after rearranging the tensor product up to isomorphism, as the homogeneouscomponent in degree j−i of the exact generalized Koszul complex (4.2) (here the differentialsare constructed using the maps `∗xj instead of r∗xj , but this doesn’t affect exactness) tensoredover k with Bj+i:

(A∗j−i ⊗B0)⊗Bj+i → · · · → (A∗1 ⊗Bj−i−1)⊗Bj+i → (A∗0 ⊗Bj−i)⊗Bj+i → 0

These rows are hence always exact except for the case i = j, since the homogeneous compo-nent of degree 0 of the exact generalized Koszul complex (4.2) starts with the term k. On theother side, the jth row (the only which is not exact) contains only the term A∗0⊗B0⊗B2j whichcan be mapped isomorphically to B2j . This mapping yields a morphism φj from Tot(D··i )to B2i which (using the exactness of rows) is easily checked to be a quasi-isomorphism. Thesum of φj over all j gives finally the desired quasi-isomorphism φ : D· → B∆.

4.4 A special truncated resolution for the diagonal

The sequence (4.3.6) providing a resolution for the diagonal is not bounded on the left.In order to apply the usual machinery to find a finite collection of generators for Db(Q), anexact truncation is necessary at a certain point of the sequence. We could just decide to stopit in some arbitrary degree M , just adding at the end of the sequence a kernel R:

0→ R→ CM → CM−1 → · · · → C1 → OX×X → O∆

but R in general has no nice description. The good point for truncating the sequence isprovided by the internal structure of the graded Clifford Algebra A, as we see in this section.

40

4.4.1 More on Clifford algebras

We first introduce the usual Clifford algebra associated to the Quadric and we discuss itsconnection with the graded Clifford algebra A defined previously.

Definition 4.4.1. Let E a vector space of dimension N = n+ 1 and let f a symmetric, nondegenerate bilinear form over E. Let A the graded Clifford k-algebra associated to f , wherethe distinguished generator in degree 2 is denoted with h. We define the (usual) Cliffordalgebra associated to f as:

Cl = Cl(E, f) =A

(h− 1)

Remark 4.4.2. The ideal (h−1) ⊂ A is not homogeneous. Hence, Cl is not a graded algebrain the classical sense. However, it has a Z/2-grading, in the sense that we can write Cl as adirect sum of modules:

Cl = Cl0 ⊕ Cl1

where, if π : A → Cl is the canonical projection, Cl0 is the image of ⊕j≥0A2j and Cl1 theimage of ⊕j≥0A2j+1. Then, the multiplication in Cl is compatible with the grading modulo2, in the sense that if x ∈ Cli and y ∈ Clj , then x · y ∈ Cli+j .

Lemma 4.4.3. For any i ≥ N − 1 = n, the multiplication by h on the right induces anisomorphism Ai → Ai+2, and the canonical projection Ai → Cli is an isomorphism.

Proof. To prove the first claim, note that, if x0, . . . , xn is an orthogonal k-basis for E ' A1,then for any i ≥ 0 (reasoning as in Lemma 4.2.6), we can find a basis for Ai given by theelements xj1 . . . xjrhp such that (1) 0 ≤ j1 < · · · < jr ≤ n and (2) r + 2p = i.

If i ≥ n + 2, these conditions force p > 0. In other words, if i ≥ n, then any element ofsuch basis for Ai+2 contains a factor hp with p > 0. Then we can define a linear map:

Ai+2 −→ Ai

xj1 . . . xjrhp 7−→ xj1 . . . xjrh

p−1

which is inverse to the map Ai → Ai+2 of multiplication by h on the right.Now we prove the second claim; thanks to the first part we can suppose i ≥ n+ 2.To prove surjectivity, notice that Cli is generated over k by the elements xJ,p = xj1 . . . xjrh

p

where J = (j1, . . . , jr) is a multi-index and (J, p) is any pair satisfying (1) j0 < j1 < · · · <jr ≤ n and (2) r+ 2p = i+ 2l for some l ≥ 0 such that i+ 2l ≥ 0. Since i ≥ n+ 2 and r ≤ n,for any such pair (J, p) we have:

2p− 2l = i− r ≥ n+ 2− n = 2p− l ≥ 1

In particular, the product xj1 . . . xjrhp−l is a well-defined element in Ai such that its imageunder the canonical projection (i.e. setting h = 1) is equal to xJ,p.

To prove injectivity, consider two elements xj1 . . . xjrhp and xj′1 . . . xj′r′hp′ in the basis

of Ai. If they have the same image in Cli, direct computation shows that necessarily r = r′,p = p′ and jl = j′l for any 0 ≤ l ≤ r.

Corollary 4.4.4. Ψi ' Ψi+2 for i ≥ N − 2 = n− 1.

41

Proof. Since the localization functor of Serre is exact, Ψi is the sheaf corresponding to thekernel of the map A∗i ⊗B → A∗i−1 ⊗B(1) in (4.3). Recall that the complex (4.3) is obtainedby the (twisted) truncation of the exact complex (4.2). Then, we can define alternativelyΨi as the sheaf corresponding to the cokernel of the differential di+2 : A∗i+2 ⊗ B(−2) →A∗i+1 ⊗B(−1).

If i ≥ N − 2, then i + 1 ≥ N − 1 and, by the lemma, we have a commutative diagramwith exact rows and vertical isomorphisms:

A∗i+2 ⊗B A∗i+1 ⊗B(1) coker(di+2) 0

A∗i+4 ⊗B A∗i+3 ⊗B(1) coker(di+4) 0

di+2

di+4

which induces an isomorphism Ψi → Ψi+2 at the level of sheaves.

Consider now the exact complex C· defined in 4.3.5, which provides the infinite resolutionof the diagonal. As a consequence of Corollary 4.4.4, when i ≥ N−2, we have an isomorphism

Ci+2 = p∗Ψi+2 ⊗ q∗OQ(−i− 2) ' p∗Ψi ⊗ q∗OQ(−i− 2) ' Ci ⊗ q∗OQ(−2)

Then, starting from N −2 and moving to the left (in the direction of increasing indexes), theobjects of the complex C· repeat themselves with a period of 2, modulo some twisting on thesecond component. This indicates heuristically a good point for truncating the resolution.

Definition 4.4.5. Denote withR the kernel of the differential CN−3 → CN−4 or, equivalentlyby exactness, the cokernel of the differential CN−1 → CN−2.

Definition 4.4.6. Define a double complex S·· of B2-modules, concentrated in the firstquadrant of cohomology:

......

. . . ⊕j(A∗N+1 ⊗Bj−2)⊗Bj−N+1 ⊕j(A∗N ⊗Bj−2)⊗Bj−N+2

. . . ⊕j(A∗N ⊗Bj−1)⊗Bj−N+1 ⊕j(A∗N−1 ⊗Bj−1)⊗Bj−N+2

where the vertical differentials are, under the usual notations,∑

i(r∗xi ⊗ `ξi) ⊗ 1 and the

horizontal ones are∑

i(`∗xi ⊗ 1) ⊗ `ξi . The structure of graded B2-modules is suggested by

the parenthesis.If the right-bottom corner is denoted with the double index (0,0), the object in position

(l,m) will be Sl,m = ⊕j(A∗N−1+m ⊗Bj−1+l)⊗Bj−2+m.

Lemma 4.4.7. Consider the double complex S·· of sheaves over Q × Q obtained by Serrelocalization of the double complex of B2-modules S··. Then the total complex S· = Tot(S··)is a left resolution for R (in the notation of Definition 4.4.5).

Proof. By exactness of Serre localization, the l-th column of the complex S·· is a left reso-lution for CN−2+l = p∗ΨN−2+l ⊗ q∗OQ(−N + 2− l). Adding the cokernels in the row -1 weobtain an augmentation of the truncated complex C≥N−2 by the double complex S··:

42

......

. . . S1,1 S1,0

. . . S0,1 S0,0

. . . CN−1 CN−2

. . . 0 0

By homological algebra we obtain a quasi-isomorphism from the total complex S· to C≥N−2,which is in its turn by definition a left resolution for R. Then we have a quasi-isomorphismS· → R proving the assertion.

4.4.2 Spinor bundles

We give now a description of the kernel R in terms of objects coming from representationtheory. We will only sketch the result, referring to [Boh05, 3.2] for more details. An alterna-tive reference for the part about representation of Clifford algebras is [FH04, Lecture 20].

Fact. For N even, the Clifford algebra Cl is simple and semi-simple. There exists anirreducible Cl-module S such that Cl ' Endk(S). The Cl-module S, seen as a Cl0-module,decomposes as S+⊕S−, and Cl0 ' End(S+)⊕End(S−); the multiplication by Cl1 in S sendsS+ to S− and viceversa. Then, two complexes of B-modules are defined:

M ·− : · · · → S∗− ⊗B(2k)→ S∗+ ⊗B(2k − 1)→ · · · → S+ ⊗B(1)→ S− ⊗BM ·+ : · · · → S∗+ ⊗B(2k)→ S∗− ⊗B(2k − 1)→ · · · → S− ⊗B(1)→ S+ ⊗B

The associated sequences of sheaves M− and M+ define two vector bundles on Q, calledspinor bundles, defined as follows. If N ≡4 0, M− and M+ are defined such that thefollowing sequences are exact:

M ·− → Σ− → 0

M ·+ → Σ+ → 0.

If N ≡4 2, M− and M+ are defined such that the following sequences are exact:

M ·+ → Σ− → 0

M ·− → Σ+ → 0.

For N odd, the Clifford algebra Cl is semi-simple and equal to the direct sum of twoisomorphic simple algebras. There is an irreducible module S over the first simple summandand an irreducible module S′ over the second simple summand, such that S ' S′ as k-modules, Cl ' End(S) ⊕ End(S′) and Cl0 ' Cl1 ' End(S). We can define a complex ofB-modules:

M · : · · · → S∗ ⊗B(−2)→ S∗ ⊗B(−1)→ S∗ ⊗B

43

The associated sequence of sheaves M define a vector bunde Σ on Q, such that the followingsequence is exact:

M · → Σ→ 0.

Proposition 4.4.8. ([Kap88, Proposition 4.7], [Boh05, Lemma 3.2.4]) The kernel R (in thenotation of Definition 4.4.5) is isomorphic to:

p∗Σ(−1)⊗ q∗Σ(−N + 2) for N oddp∗Σ+(−1)⊗ q∗Σ+(−N + 2)⊕ p∗Σ−(−1)⊗ q∗Σ−(−N + 2) for N ≡4 2

p∗Σ+(−1)⊗ q∗Σ−(−N + 2)⊕ p∗Σ−(−1)⊗ q∗Σ+(−N + 2) for N ≡4 0

Proof. (Sketch). We show the proof for the case of N odd. We have

Cl0 ' Cl1 ' End(S) '' S∗ ⊗ S

Cl∗0 ' Cl∗1 ' S ⊗ S

∗.

Then, by Lemma 4.4.3, we have that for any i:

A∗i ' Cli ' S ⊗ S∗.

Substituting A∗i in the double complex (4.4.6) and rearranging the tensor products, we obtain

......

. . . ⊕j(S∗ ⊗Bj−2)⊗ (S ⊗Bj−N+1) ⊕j(S∗ ⊗Bj−2)⊗ (S ⊗Bj−N+2)

. . . ⊕j(S∗ ⊗Bj−1)⊗ (S ⊗Bj−N+1) ⊕j(S∗ ⊗Bj−1)⊗ (S ⊗Bj−N+2)

The associated double complex of sheaves on Q × Q results to be isomorphic to the doublecomplex arising from the tensor product of p∗M ·(−1) and q∗M ·(−N + 2). Then, we havean isomorphism between the total complexes:

R ' p∗M ·(−1)⊗ q∗M ·(−N + 2)

Finally, we have a quasi-isomorphism M · ' Σ, that gives a quasi-isomorphism:

p∗M ·(−1)⊗ q∗M ·(−N + 2) ' p∗Σ(−1)⊗ q∗Σ(−N + 2)

We conclude that

R ' p∗Σ(−1)⊗ q∗Σ(−N + 2).

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4.5 The derived category Db(Q)

We are now ready to give a description of the derived category Db(Q). First, we use thenotation Σ(±) meaning that for N even both Σ+ and Σ− are considered, while for N evenonly Σ is. We have two distinguished families of sheaves on Q:

X = Σ(±)(−1),ΨN−3, . . . ,Ψ0 = OQY = Σ(±)(−N + 2),OQ(−N + 3), . . . ,OQ

Both X and Y generate Db(Q) as a triangulated category. We then use now the followingfact ([Kap88, Proposition 4.9]):

Lemma 4.5.1. For each pair of bundles E and F of the set X (respectively Y ), ExtmOG(E,F ) =0 for all m > 0 and HomOG(E,E) = k.

This lemma implies that both X and Y are strong exceptional sets. Indeed, for any Eand F in X (resp. in Y ),

HomDb(Q)(E,F [m]) = ExtmDb(Q)(E,F ) = ExtmOG(E,F ).

Then, recalling that classical Ext groups are always zero in negative degrees, we obtain forE = F

HomDb(Q)(E,E[m]) =

0 for m < 0

0 for m > 0

k for m = 0

and, in general, HomDb(Q)(E,F [m]) = 0 for anym 6= 0. Then, we can state our final theorem:

Theorem 4.5.2. The derived category Db(Q) is equivalent as a triangulated category toKb(X) (resp. to Kb(Y )) i.e. the homotopy category of finite complexes of sheaves whoseterms are finite direct sums of the elements of X (respectively, of Y ).

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Bibliography

[AO89] V. Ancona and G. Ottaviani. An introduction to the derived categories and thetheorem of beilinson. Atti Accademia Peloritana dei Pericolanti, cl. I, 67, 1989.

[Bei78] A. A. Beilinson. Coherent sheaves on Pn and problems of linear algebra. FunctionalAnalysis and Its Applications, 12, 1978.

[BGS78] I. N. Bernshtein, I. M. Gel’fand, and Gel’fand S.I. Algebraic vector bundles on Pnand problems of linear algebra. Functional Analysis and Its Applications, 12, 1978.

[Boh05] C. Bohning. Derived categories of coherent sheaves on rational homogeneous man-ifolds. arXiv:math/0506429 [math.AG], 2005. Available at http://arxiv.org/abs/math/0506429v1.

[Cal05] A. Caldararu. Derived categories of sheaves: a skimming. arXiv:math/0501094[math.AG], 2005. Available at http://arxiv.org/abs/math/0501094.

[Cal09] J. R. Calabrese. On a theorem by beilinson. Master’s thesis, Università di Pisa,2009. Available at http://math.rice.edu/~jrc9/stuff/beilinson.pdf.

[EF05] D. Edidin and C. Francisco. Grassmannians and representations.arXiv:math/0507482 [math.AG], 2005. Available at http://arxiv.org/abs/math/0507482v2.

[EH16] D. Eisenbud and J. Harris. 3264 and All That. A Second Course in AlgebraicGeometry. Cambridge University Press, 2016. Graduate Texts in Mathematics.

[FH04] W. Fulton and J. Harris. Representation Theory. A First Course. Springer-VerlagNew York, 2004.

[FL85] W. Fulton and S. Lang. Riemann-Roch Algebra. Springer-Verlag, 1985.

[Gro57a] A. Groethendieck. Sur quelques points d’algèbre homogique. Tôhoku MathematicalJournal, (2) 9, 1957.

[Gro57b] A. Groethendieck. Théorèmes de dualité pour les faisceaux algeébriques cohérents.In Seminaire Bourbaki 149. Secretariat Mathematique IHP, 1957.

[Gro58] A. Groethendieck. The cohomology theory of abstract algebraic varieties. In In-ternational Congress of Mathematics at Edinburgh. Cambridge University Press(1960), 1958.

[Gro63] A. Grothendieck. Étude cohomologique des faisceaux cohérents, seconde partie.(ega iii.2). Publications Mathématiques de l’IHÉS, 17, 1963.

46

[Har66] R. Hartshorne. Residues and Duality. Springer-Verlag, 1966.

[Har77] R. Hartshorne. Algebraic Geometry. Springer, 1977.

[Huy06] D. Huybrechts. Fourier-Mukai Transforms in Algebraic Geometry. Clarendon Press,2006. Oxford Mathematical Monographs.

[Kap85] M. M. Kapranov. On the derived categories of coherent sheaves on grassman man-ifolds. Mathematics of the USSR-Izvestiya, 24.1, 1985.

[Kap86] M. M. Kapranov. On the derived categories of coherent sheaves on the square.Functional Analysis and Its Applications, 20.2, 1986.

[Kap88] M. M. Kapranov. On the derived categories of coherent sheaves on some homoge-neous spaces. Inventiones mathematicae, 92, 1988.

[Kem93] G. Kempf. Algebraic Varieties. Cambridge University Press, 1993. London Math-ematical Society Lecture Note Series 172.

[Kol04] R. Kolhatkar. Grassmann varieties. Master’s thesis, Department of Mathematicsand Statistics, McGill University, Montreal, 2004. Available at http://www.math.mcgill.ca/goren/Students/KolhatkarThesis.pdf.

[Lur] J. Lurie. A proof of the borel–weil–bott theorem. Available at http://www.math.harvard.edu/~lurie/papers/bwb.pdf.

[nLa] nLab. Schur functor. Available at https://ncatlab.org/nlab/show/Schur+functor. Visited: May 2016.

[PP05] A. Polishchuk and L. Positselski. Quadratic algebras, volume 37. American Math-ematical Society, 2005. University Lecture Series.

[Swa85] R. Swan. K-theory of quadric hypersurfaces. Annals of Mathematics, 122, 1985.

[Tat57] J. Tate. Homology of noetherian rings and local rings. Illinois J. Math., 1(1), 031957.

[Ver96] J.-L. Verdier. Des catégories dérivées des catégories abéliennes. Astérisque, 239,1996.

[Wei94] C. A. Weibel. An Introduction to homological algebra. Cambridge University Press,1994.

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