RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES

GONCALO TABUADA

Abstract. Written for the proceedings of the second Mid-Atlantic Topol-

ogy Conference, held at Johns Hopkins University, this survey covers some

of the recent developments on noncommutative motives and their applica-tions. Among other topics, we compute the additive invariants (e.g. algebraic

K-theory, cyclic homology, topological Hochschild homology, etc) of relative

cellular spaces and orbifolds; describe some of the equivariant counterpartsof the theory of noncommutative pure motives; prove Voevodsky’s nilpotence

conjecture, as well as Grothendieck’s standard conjectures of type C+ and D,

in some new cases; embed the Brauer group into secondary K-theory; describethe behavior of the localizing A1-invariants (e.g. homotopy K-theory, etale

K-theory, periodic cyclic homology, etc) with respect to open/closed schemedecompositions and corner skew Laurent polynomial algebras; relate Kontse-

vich’s category of noncommutative mixed motives to Morel-Voevodsky’s stable

A1-homotopy category, to Voevodsky’s triangulated category of motives, andto Levine’s triangulated category of mixed motives; prove Kimura-finiteness for

quadric fibrations over smooth curves; and finally explain how Grothendieck’s

theory of periods can be extended to the noncommutative world.

To Lily, for being by my side.

Introduction

After the release of the monograph [Noncommutative Motives. With a prefaceby Yuri I. Manin. University Lecture Series 63, American Mathematical Society,2015], several important results on the theory of noncommutative motives havebeen established. The purpose of this survey, written for a broad mathematicalaudience, is to give a rigorous overview of some of these recent results. We willfollow closely the notations, as well as the writing style, of the monograph [52].Therefore, we suggest the reader to have it at his/her desk while reading thissurvey. The monograph [52] is divided into the following chapters:

Chapter 1. Differential graded categories.Chapter 2. Additive invariants.Chapter 3. Background on pure motives.Chapter 4. Noncommutative pure motives.Chapter 5. Noncommutative (standard) conjectures.Chapter 6. Noncommutative motivic Galois groups.Chapter 7. Jacobians of noncommutative Chow motives.Chapter 8. Localizing invariants.Chapter 9. Noncommutative mixed motives.Chapter 10. Noncommutative motivic Hopf dg algebrasAppendix A. Grothendieck derivators.

Date: December 12, 2016.The author is very grateful to the organizers Nitu Kitchloo, Mona Merling, Jack Morava, Emily

Riehl, and W. Stephen Wilson, for the kind invitation to present some of this work at the secondMid-Atlantic Topology Conference. The author was supported by a NSF CAREER Award.

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2 GONCALO TABUADA

In this survey we cover some of the recent developments concerning the Chapters2, 4, 5, 8, and 9. These developments are described below in Sections 1, 2, 3, 4,and 5, respectively. The final Section 6, entitled “Periods of differential gradedcategories”, discusses a recent research subject which was not addressed in [52].

Preliminaries. Throughout the survey, k will denote a base field. We will assumethe reader is familiar with the language of differential graded (=dg) categories; for asurvey on dg categories, we invite the reader to consult Keller’s ICM address [27]. Inparticular, we will freely use the notions of Morita equivalence of dg categories (see[52, §1.6]) and smooth/proper dg category in the sense of Kontsevich (see [52, §1.7]).We will write dgcat(k) for the category of (small) dg categories and dgcatsp(k) forthe full subcategory of smooth proper dg categories. Given a k-scheme X (or moregenerally an algebraic stack X ), we will denote by perfdg(X) the canonical dgenhancement of the category of perfect complexes perf(X); see [52, Example 1.27].

1. Additive invariants

Recall from [52, §2.3] the construction of the universal additive invariant of dgcategories U : dgcat(k) → Hmo0(k). In [52, §2.4], we described the behavior of Uwith respect to semi-orthogonal decompositions, full exceptional collections, purelyinseparable field extensions, central simple algebras, sheaves of Azumaya algebras,twisted flag varieties, nilpotent ideals, finite dimensional algebras of finite globaldimension, etc. In §1.1-1.2 below, we describe the behavior of U with respect torelative cellular spaces and (global) orbifolds. As explained in [52, Thm. 2.9], allthe results of this section are motivic in the sense that they hold similarly for everyadditive invariant such as algebraic K-theory, mod-lν algebraic K-theory, Karoubi-Villamayor K-theory, nonconnective algebraic K-theory, homotopy K-theory, etaleK-theory, Hochschild homology, cyclic homology, negative cyclic homology, periodiccyclic homology, topological Hochschild homology, topological cyclic homology, etc.

In order to simplify the exposition, given a k-scheme X (or more generally analgebraic stack X ), we will write U(X) instead of U(perfdg(X)).

1.1. Relative cellular spaces. A flat morphism of k-schemes p : X → Y is calledan affine fibration of relative dimension d if for every point y ∈ Y there existsa Zariski open neighborhood y ∈ V such that XV := p−1(V ) ' Y × Ad withpV : XV → Y isomorphic to the projection onto the first factor. Following Karpenko[25, Def. 6.1], a smooth projective k-scheme X is called a relative cellular space ifthere exists a filtration by closed subschemes

∅ = X−1 ↪→ X0 ↪→ · · · ↪→ Xi ↪→ · · · ↪→ Xn−1 ↪→ Xn = X

and affine fibrations pi : Xi\Xi−1 → Yi, 0 ≤ i ≤ n, of relative dimension di with Yia smooth projective k-scheme. The smooth k-schemes Xi\Xi−1 are called the cellsand the smooth projective k-schemes Yi the bases of the cells.

Example 1.1 (Gm-schemes). The celebrated Bialynicki-Birula decomposition [6]provides a relative cellular space structure on smooth projective k-schemes equippedwith a Gm-action. In this case, the bases of the cells are given by the connectedcomponents of the fixed point locus. This class of relative cellular spaces includesthe isotropic flag varieties considered by Karpenko in [25] as well as the isotropichomogeneous spaces considered by Chernousov-Gille-Merkurjev in [13].

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 3

Theorem 1.2 ([61, Thm. 2.7]). For every relative cellular space X, we have aninduced isomorphism U(X) '

⊕ni=0 U(Yi).

Theorem 1.2 shows that the additive invariants of relative cellular spaces Xare completely determined by the basis Yi of the cells Xi\Xi−1. Its proof makesessential use of Theorem 4.6 below; consult [61, §9] for details.

Example 1.3 (Knorrer periodicity). Let q = fg+q′, where f , g, and q′, are forms ofdegree a > 0, b > 0, and a+ b, in disjoint sets of variables (xi)i=1,...,m, (yj)j=1,...,n,and (zk)k=1,...,p, respectively. Such a decomposition holds, in particular, for allisotropic quadratic forms q. Let us write Q and Q′ for the projective hypersurfacesdefined by q and q′, respectively. Assume that Q is smooth. Under this assumption,we have a Gm-action on Q given by λ · (x, y, z) := (λbx, λ−ay, z) with fixed point

locus Pm−1qPn−1qQ′; this implies that Q′ is also smooth. By combining Theorem1.2 and Example 1.1 with the fact that U(Pn) ' U(k)⊕(n+1) (see [52, §2.4.2]), weobtain an induced isomorphism U(Q) ' U(k)⊕(m+n) ⊕ U(Q′). Morally speaking,this shows that modulo the additive invariants of the base field k, the correspondingadditive invariants of Q and Q′ are the same.

1.2. Orbifolds. Let G be a finite group of order n (we assume that 1/n ∈ k), ϕthe set of all cyclic subgroups of G, and ϕ/∼ a (chosen) set of representatives ofthe conjugacy classes in ϕ. Let X be a smooth separable k-scheme equipped witha G-action (=G-scheme) and [X/G] the associated (global) orbifold.

As explained in [62, §3], the assignment [V ] 7→ V ⊗k −, where V stands for a G-representation, gives rise to an action of the representation ring R(G) on U([X/G]).Given σ ∈ ϕ, let eσ be the (unique) idempotent of the Z[1/n]-linearized representa-tion ring R(σ)1/n whose image under all the restrictions R(σ)1/n → R(σ′)1/n, σ

′ (σ, is zero. In what follows, we write R(σ)1/n instead of eσR(σ)1/n and U([Xσ/σ])1/n

instead of eσU([Xσ/σ])1/n. Note that the normalizerN(σ) acts naturally on [Xσ/σ]

and hence on U([Xσ/σ])1/n. By functoriality, this action restricts to U([Xσ/σ])1/n.

Theorem 1.4 ([62, Thm. 1.1 and Cor. 1.6(i)]). We have an induced isomorphism

(1.5) U([X/G])1/n '⊕σ∈ϕ/∼

U([Xσ/σ])N(σ)1/n .

If k contains the nth roots of unity, then (1.5) reduces furthermore to

(1.6) U([X/G])1/n '⊕σ∈ϕ/∼

(U(Xσ)1/n ⊗Z[1/n] R(σ)1/n)N(σ) ,

where −⊗Z[1/n]− stands for the canonical action of the category of finitely generatedprojective Z[1/n]-modules on Hmo0(k)1/n.

Roughly speaking, the above formula (1.6) shows that the additive invariants of(global) orbifolds can be computed using solely “ordinary” schemes. This is theconceptual idea behind the proof of Theorem 3.19 below.

Example 1.7 (McKay correspondence). In many cases, the dg category perfdg([X/G])is known to be Morita equivalent to perfdg(Y ) for a crepant resolution Y of the(singular) geometric quotient X//G; see [5, 12, 24, 26]. This is generally referred toas the “McKay correspondence”. Whenever it holds, we can replace [X/G] by Y inthe above formulas (1.5)-(1.6). Here is an illustrative example (with k algebraically

4 GONCALO TABUADA

closed): the cyclic group G = C2 acts on any abelian surface S by the involutiona 7→ −a and the Kummer surface Km(S) is defined as the blow-up of S//C2 in its16 singular points. In this case, the dg category perfdg([S/C2]) is Morita equivalentto perfdg(Km(S)). Consequently, Theorem 1.4 leads to an isomorphism

(1.8) U(Km(S))1/2 ' U(S)C2

1/2 ⊕ U(k)⊕161/2 .

Since the Kummer surface is Calabi-Yau, the category perf(Km(S)) does not admitany non-trivial semi-orthogonal decompositions. Therefore, the above decomposi-tion (1.8) is not induced from a semi-orthogonal decomposition.

1.2.1. Algebraic K-theory. As mentioned above, algebraic K-theory is an additiveinvariant. Therefore, in the case where k contains the nth roots of unity, Theorem1.4 leads to the following isomorphism

(1.9) K∗([X/G])1/n '⊕σ∈ϕ/∼

(K∗(Xσ)1/n ⊗Z[1/n] R(σ)1/n)N(σ) .

The formula (1.9) was originally established by Vistoli in [65, Thm. 1]. Amongother ingredients, Vistoli’s proof makes essential use of devissage. The proof ofTheorem 1.4, and consequently of the above formula (1.9), is not only different butmoreover avoids the use of devissage; consult [62, §6] for details.

1.2.2. Inertia formulas. Let E : dgcat(k)→ D be an additive invariant. As provedin [62, Cor. 1.6(ii)], if k contains the nth roots of unity and the category D is l-linear for a field l which contains the nth roots of unity and 1/n ∈ l, then the aboveisomorphism (1.6) reduces furthermore to the following “inertia” formula(s)

(1.10) E([X/G]) '⊕g∈G/∼

E(Xg)C(g) ' (⊕g∈G

E(Xg))G ,

where C(g) stands for the centralizer of g. This applies notably to Hochschildhomology, cyclic homology, negative cyclic homology, periodic cyclic homology,and also to topological Hochschild homology; consult [62, Examples 1.12 and 1.16].Moreover, in all these examples the assumption that k contains the nth roots of unitycan be removed. In the case of cyclic homology and its variants, these “inertia”formulas were originally established by Baranovsky in [3, Thm. 1.1]. Baranovsky’sproof is very specific to cyclic homology. In constrast, the proof of Theorem 1.4, andhence of (1.10), avoids all the specificities of cyclic homology and is quite conceptual.Finally, to the best of the author’s knowledge, in the case of topological Hochschildhomology the above “inertia” formula (1.10) is new in the literature.

1.2.3. Twisted versions. Given a sheaf of Azumaya algebras F over [X/G], i.e. aG-equivariant sheaf of Azumaya algebras over X, all the above formulas admitsa F-twisted version; consult [62, §1] for details. For example, in the particularcase where k = C and F is induced from a cohomology class [α] ∈ H2(G,C×) bypull-back along the morphism [X/G]→ [•/G], the α-twisted version of (1.9) is thealgebraic analogue of Adem-Ruan’s formula for twisted orbifold K-theory [1, §1].

2. Noncommutative pure motives

The category of noncommutative Chow motives NChow(k) is defined as the idem-potent completion of the full subcategory of Hmo0(k) consisting of the objectsU(A), with A a smooth proper dg category; see [52, §4.1]. By construction, this

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 5

category is additive, rigid symmetric monoidal, and comes equipped with a sym-metric monoidal functor U : dgcatsp(k) → NChow(k). Moreover, we have naturalisomorphisms HomNChow(k)(U(A), U(B)) ' K0(Aop ⊗ B).

Given a rigid symmetric monoidal category (C,⊗,1), consider the ⊗-ideal

⊗nil(a, b) := {f ∈ HomC(a, b) | f⊗n = 0 for some n� 0} .

Recall from [52, §4.4] that the category of noncommutative ⊗-nilpotent motivesNVoev(k) is defined as the idempotent completion of the quotient NChow(k)/⊗nil.

As explained in [52, §4.5], periodic cyclic homology gives rise to an additivesymmetric monoidal functor HP± : NChow(k) → VectZ/2(k), with values in thecategory of finite dimensional Z/2-graded k-vector spaces. Recall from loc. cit.that the category of noncommutative homological motives NHom(k) is defined asthe idempotent completion of the quotient NChow(k)/Ker(HP±).

Given a rigid symmetric monoidal category (C,⊗,1), consider the ⊗-ideal

N (a, b) := {f ∈ HomC(a, b) | ∀g ∈ HomC(b, a) we have tr(g ◦ f) = 0} ,

where tr(g ◦ f) stands for the categorical trace of the endomorphism g ◦ f . Recallfrom [52, §4.6] that the category of noncommutative numerical motives NNum(k)is defined as the idempotent completion of the quotient NChow(k)/N .

In §2.1 below, we describe a precise relation between the Brauer group of k andthe (Grothendieck ring of the) category of noncommutative Chow motives. In §2.2,given a finite group G, we describe some of the G-equivariant counterparts of theaforementioned theory of noncommutative pure motives.

2.1. Brauer group. Let Br(k) be the Brauer group of the base field k. Given acentral simple k-algebra A, we write [A] for the corresponding Brauer class.

Example 2.1 (Local fields). A local field k is isomorphic to R, to C, to a finite fieldextension of Qp, or to a finite field extension of Fp((t)). Thanks to local class fieldtheory, we have Br(R) ' Z/2, Br(C) = 0, and Br(k) ' Q/Z in all the remainingcases. Moreover, every element of Br(k) can be represented by a cyclic k-algebra.

Recall from [52, §2.4.4] that we have the following equivalence

(2.2) [A] = [B]⇔ U(A) ' U(B)

for any two central simple k-algebras A and B. Intuitively speaking, (2.2) showsthat the Brauer class [A] and the noncommutative Chow motive U(A) containexactly the same information. Let us denote by K0(NChow(k)) the Grothendieckring of the additive symmetric monoidal category NChow(k). Given a central simplek-algebra A, we write [U(A)] for the Grothendieck class of U(A). The (proof ofthe) next result is implicitly contained in [51, Thm. 6.12][55, Thm. 1.3]:

Theorem 2.3. Given central simple k-algebras A and B, we have the equivalence:

(2.4) U(A) ' U(B)⇔ [U(A)] ' [U(B)] .

Roughly speaking, Theorem 2.3 shows that the noncommutative Chow motivesof central simple k-algebras are insensitive to the Grothendieck group relations.Among other ingredients, its proof makes use of the category of noncommutativenumerical motives; consult [51, §6][55, §3] for details. Note that by combining theabove equivalences (2.2) and (2.4), we obtain the following result:

6 GONCALO TABUADA

Corollary 2.5. The following map is injective

Br(k) −→ K0(NChow(k)) [A] 7→ [U(A)] .

Consult §4.1.1 for the implications of Corollary 2.5 to secondary K-theory.

Remark 2.6 (Generalizations). Theorem 2.3 and Corollary 2.5 hold more generallywith k replaced by a (certain) base k-scheme X. Furthermore, instead of the Brauergroup Br(k), we can consider the second etale cohomology group1 H2

et(X,Gm);consult [51, 55] for details. Recall, for example, that in the case of the cone over asmooth irreducible plane complex curve of degree ≥ 4, the latter etale cohomologygroup contains non-torsion classes. The same phenomenon occurs in the case ofMumford’s (celebrated) singular surface [47, page 75]; see [55, Example 1.32].

Remark 2.7 (Jacques Tits’ motivic measure). The Grothendieck ring of varietiesK0Var(k), introduced in a letter from Grothendieck to Serre in the sixties, is definedas the quotient of the free abelian group on the set of isomorphism classes of k-schemes by the “cut-and-paste” relations. Although very important, the structureof this ring still remains poorly understood. Among other ingredients, the aboveTheorem 2.3 was used in the construction of a new motivic measure µT entitledTits motivic measure; consult [59] for details. This motivic measure led to the proofof several new structural properties of K0Var(k). For example, making use of µT , itwas proved in loc. cit. that two quadric hypersurfaces (or more generally involutionvarieties), associated to quadratic forms of degree 6, have the same Grothendieckclass if and only if they are isomorphic. In the same vein, two products of conicshave the same Grothendieck class if and only if they are isomorphic; this refines aprevious result of Kollar [32].

2.2. Equivariant noncommutative motives. Let G be a finite group such thatchar(k) - |G|. Recall from [53, Def. 4.1] that a G-action on a dg category A, denotedby G � A, consists of the following data:

(i) a family of dg equivalences φσ : A → A, σ ∈ G, with φ1 = id;(ii) a family of natural isomorphisms of dg functors ερ,σ : φρ ◦ φσ ⇒ φρσ, σ, ρ ∈ G,

with ε1,σ = εσ,1 = id, such that the equality ετρ,σ◦(ετ,ρ◦φσ) = ετ,ρσ◦(φτ ◦ερ,σ)holds for every σ, ρ, τ ∈ G.

Recall from [53, Def. 4.17] that a G-equivariant object in G � A consists of an objectx ∈ A and of a family of closed degree zero isomorphisms θσ : x → φσ(x), σ ∈ G,with θ1 = id, such that the equality θρ ◦ φρ(θσ) ◦ ερ,σ(x) = θρσ holds for everyσ, ρ ∈ G. Let us denote AG the associated dg category of G-equivariant objects.From a topological viewpoint, AG may be understood as the “homotopy fixedpoints” of the G-action on A. Here are four illustrative families of examples:

Example 2.8 (G-schemes). Given a G-scheme X, the dg category perfdg(X) inher-its a G-action induced by the pull-back dg equivalences φσ := σ∗. In this case,the associated dg category of G-equivariant objects is Morita equivalent to the dgcategory of G-equivariant perfect complexes perfGdg(X) = perfdg([X/G]).

Example 2.9 (Line bundles). Let X be a k-scheme. If G can be realized as asubgroup of the Picard group Pic(X), then the dg category perfdg(X) inherits aG-action induced by the dg equivalences φσ := − ⊗OX Lσ, where Lσ stands for

1As proved by Gabber [16] and de Jong [22], in the case where X admits an ample line bundle(e.g. X affine), the Brauer group Br(k) may be identified with the torsion subgroup of H2

et(X,Gm).

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 7

the invertible line bundle corresponding to σ ∈ G. In this case, the associated dgcategory of G-equivariant objects is Morita equivalent to perfdg(Y ), where Y :=

SpecX(⊕

σ∈G L−1σ ) stands for a non-ramified |G|-fold cover of X.

Example 2.10 (G-algebras). A G-action on a k-algebra A is equivalent to a G-actionas above with ερ,σ = id. In this case, the associated dg category of G-equivariantobjects is Morita equivalent to the semi-direct product k-algebra AoG.

Example 2.11 (Cohomology classes). Given a cohomology class [α] ∈ H2(G, k×),the dg category k inherits a G-action, denoted by G �α k, with φσ := id andερ,σ := α(ρ, σ). In this case, the associated dg category of G-equivariant objects isMorita equivalent to the twisted group algebra kα[G].

Let dgcatG(k) be the category of (small) dg categories equipped with a G-action,

and dgcatGsp(k) the full subcategory of smooth proper dg categories equipped with aG-action. As explained in [53, §5], the category of noncommutative Chow motives

NChow(k) admits a G-equivariant counterpart NChowG(k). Recall from loc. cit.that the latter category is additive, rigid symmetric monoidal, and comes equippedwith a symmetric monoidal functor UG : dgcatG(k) → NChow(k). Moreover, wehave the following natural isomorphisms

HomNChowG(k)(UG(G � A), UG(G � B)) ' KG

0 (Aop ⊗ B) ,

where the right-hand side stands for the G-equivariant Grothendieck group. Inparticular, the ring of endomorphisms of the ⊗-unit UG(G �1 k) agrees with therepresentation ring Rk(G) of the group G. For example, when G is abelian, RC(R)

is isomorphic to the group ring Z[G] of the character group G. Let us write I forthe (augmentation) ideal associated to the rank homomorphism Rk(G) � Z.

Remark 2.12. The categories of noncommutative ⊗-nilpotent motives, noncommu-tative homological motives, and noncommutative numerical motives, also admitG-equivariant counterparts; consult [53, §5.4] for details.

2.2.1. Relation with equivariant Chow motives. Making use of Edidin-Graham’swork [15] on equivariant intersection theory, Laterveer [42], and Iyer and Muller-Stach [21], extended the theory of Chow motives to the G-equivariant setting. In

particular, they constructed a category of G-equivariant Chow motives ChowG(k)

and a (contravariant) symmetric monoidal functor hG : SmProjG(k)→ ChowG(k),defined on smooth projective G-schemes.

Theorem 2.13 ([53, Thm. 8.4]). There exists a Q-linear, fully-faithful, symmetricmonoidal ΦGQ making the following diagram commute

SmProjG(k)X 7→G�perfdg(X)

//

hG(−)Q��

dgcatGsp(k)

UG(−)Q��ChowG(k)Q

��

NChowG(k)Q

(−)IQ��ChowG(k)Q/−⊗Q(1)

ΦGQ

// NChowG(k)Q,IQ ,

8 GONCALO TABUADA

where ChowG(k)Q/−⊗Q(1) stands for the orbit category of ChowG(k)Q with respectto the G-equivariant Tate motive Q(1) (see [52, §4.2]), and (−)IQ for the localizationfunctor associated to the augmentation ideal IQ.

Roughly speaking, Theorem 2.13 shows that in order to compare the equivari-ant commutative world with the equivariant noncommutative world, we need to“⊗-trivialize” the G-equivariant Tate motive Q(1) on one side and to localize atthe augmentation ideal IQ on the other side. Only after these two reductions,the equivariant commutative world embeds fully-faithfully into the equivariantnoncommutative world. As illustrated in §2.2.2 below, this shows that the G-equivariant Chow motive hG(X)Q and the G-equivariant noncommutative Chowmotive UG(G � perfdg(X)) contain (important) independent information about X.

2.2.2. Full exceptional collections. Let X be a smooth projective G-scheme. Inorder to study it, we can proceed into two distinct directions. On one direction, wecan associate to X its G-equivariant Chow motive hG(X)Q. On another direction,we can associate to X its G-category of perfect complexes G � perf(X). Thefollowing result, whose proof makes essential use of the above Theorem 2.13, relatesthese two distinct directions of study:

Theorem 2.14 ([53, Thm. 1.2]). If the category perf(X) admits a full exceptionalcollection (E1, . . . , En) of G-invariant objects2, i.e. σ∗(Ei) ' Ei for every σ ∈ G,then there exists a choice of integers r1, . . . , rn ∈ {0, . . . ,dim(X)} such that

(2.15) hG(X)Q ' L⊗r1 ⊕ · · · ⊕ L⊗rn ,

where L ∈ ChowG(k)Q stands for the G-equivariant Lefschetz motive.

Theorem 2.14 can be applied, for example, to any G-action on projective spaces,quadrics, Grassmannians, etc; consult [53, Examples 9.9-9.11] for details. Morallyspeaking, Theorem 2.14 shows that the existence of a full exceptional collection ofG-invariant objects completely determines the G-equivariant Chow motive hG(X)Q.In particular, hG(X)Q loses all the information about the G-action on X. In con-trast, as explained in [53, Rmk. 9.4 and Prop. 9.8], theG-invariant objects E1, . . . , Enyield (non-trivial) cohomology classes [α1], . . . , [αn] ∈ H2(G, k×) such that

(2.16) UG(G � perfdg(X)) ' UG(G �α1k)⊕ · · · ⊕ UG(G �αn k) .

Taking into account (2.15)-(2.16), the G-equivariant Chow motive hG(X)Q andthe G-equivariant noncommutative Chow motive UG(G � perfdg(X)) should beconsidered as complementary. While the former keeps track of the Tate twists butnot of the G-action, the latter keeps track of the G-action but not of the Tate twists.

Remark 2.17. The particular case of Theorem 2.13, resp. Theorem 2.14, with Gthe trivial group, is described in [52, Thm. 4.3], resp. in [52, Thm. 4.11]. Note thatin this case the localization functor (−)IQ reduces to the identity.

3. Noncommutative (standard) conjectures

In this section we assume that k is of characteristic zero. Given a smoothproper k-scheme X and a Weil cohomology theory H∗, let πnX be the nth Kunnethprojector of H∗(X), Z∗(X)Q the Q-vector space of algebraic cycles on X, andZ∗(X)Q/∼nil, Z

∗(X)Q/∼hom, and Z∗(X)Q/∼num, the quotients with respect to the

2A G-equivariant object is G-invariant, but the converse does not holds!

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 9

smash-nilpotence, homological, and numerical equivalence relations. Recall from[52, §3.0.8-3.0.11] that the Grothendieck’s standard conjecture3 of type C+, denotedby C+(X), asserts that the even Kunneth projector π+

X :=∑n π

2nX is algebraic.

The Grothendieck’s standard conjecture4 of type D, denoted by D(X), asserts thatZ∗(X)Q/∼hom = Z∗(X)Q/∼num. The Voevodsky’s nilpotence conjecture5, denotedby V (X), asserts that Z∗(X)Q/∼nil = Z∗(X)Q/∼num. Finally, the conjecture S(X)asserts that the Chow motive h(X)Q is Schur-finite in the sense of Deligne [14, §1].

Remark 3.1 (Current status). (i) Thanks to the work of Grothendieck and Kleiman(see [17, 30, 31]), the conjecture C+(X) holds when X is of dimension ≤ 2, andalso for abelian varieties. Moreover, this conjecture is stable under products;

(ii) Thanks to the work of Lieberman [44], the conjecture D(X) holds when X isof dimension ≤ 4, and also for abelian varieties;

(iii) Thanks to the work Voevodsky [68] and Voisin [69], the conjecture V (X) holdswhen X is of dimension ≤ 2. Moreover, thanks to the work of Kahn-Sebastian[23], the conjecture V (X) also holds for abelian 3-folds;

(iv) Thanks to the work of Kimura [29] and Shermenev [50], the conjecture S(X)holds when X is of dimension ≤ 1, and also for abelian varieties.

In §3.1 below, we introduce the noncommutative counterparts of the aforemen-tioned important conjectures. Moreover, we explain how these noncommutativecounterparts enable the proof of the aforementioned conjectures in some new cases.

3.1. Noncommutative counterparts and their applications. Recall from §2that periodic cyclic homology descends to the category of noncommutative Chowmotives yielding a functor HP± : NChow(k)Q → VectZ/2(k). Given a smoothproper dg category A, consider the following Q-vector spaces

K0(A)Q/∼? := Hom?(U(k)Q, U(A)Q) ,

where ? belongs to the sets {nil,hom,num} and {NVoev(k),NHom(k)Q,NNum(k)Q},respectively. Under these notations, the aforementioned important conjecturesC+, D, V, S admit the following noncommutative counterparts:

Conjecture C+nc(A): The even Kunneth projector π+

A of HP±(A) is algebraic,

i.e. there exists an endomorphism π+A of U(A)Q such that HP±(π+

A) = π+A.

Conjecture Dnc(A): The equality K0(A)Q/∼hom = K0(A)Q/∼num holds.

Conjecture Vnc(A): The equality K0(A)Q/∼nil = K0(A)Q/∼num holds.

Conjecture Snc(A): The noncommutative Chow motive U(A)Q is Schur-finite.

Theorem 3.2. Given a smooth proper k-scheme X, we have the equivalences:

C+(X) ⇔ C+nc(perfdg(X))(3.3)

D(X) ⇔ Dnc(perfdg(X))(3.4)

V (X) ⇔ Vnc(perfdg(X))(3.5)

S(X) ⇔ Snc(perfdg(X)) .(3.6)

3The standard conjecture of type C+ is also known as the sign conjecture. If the even Kunneth

projector π+X is algebraic, then the odd Kunneth projector π−

X :=∑

n π2n+1X is also algebraic.

4As explained by Grothendieck in [17], the Weil conjectures on ζ-functions follow (when k is

of characteristic zero) from the standard conjecture of type D.5Voevodsky’s nilpotence conjecture implies Grothendieck’s standard conjecture of type D.

10 GONCALO TABUADA

Morally speaking, Theorem 3.2 shows that the original conjectures C+, D, V, Sbelong not only to the realm of algebraic geometry, but also to the broad settingof smooth proper dg categories. Consult [52, §5], and the references therein, forthe implications⇒ in (3.3)-(3.4) and for the equivalences (3.5)-(3.6). The converseimplications ⇐ in (3.3)-(3.4) were established in [56, Thm. 1.1]. In §3.1.1 below,we explain how the noncommutative viewpoint provided by Theorem 3.2 enablesthe proof of the original conjectures in several new cases. In §3.1.2, making use ofTheorem 3.2, we extend the original conjectures to smooth proper algebraic stacks.

3.1.1. Homological projective duality. For a survey on homological projective du-ality (=HPD), we invite the reader to consult Kuznetsov’s ICM address [37]. LetX be a smooth projective k-scheme equipped with a line bundle OX(1); we writeX → P(V ) for the associated morphism where V := H0(X,OX(1))∗. Assume thatthe category perf(X) admits a Lefschetz decomposition 〈A0,A1(1), . . . ,Ai−1(i−1)〉with respect to OX(1) in the sense of [38, Def. 4.1]. Following [38, Def. 6.1], let Y bethe HP-dual of X, OY (1) the HP-dual line bundle, and Y → P(V ∗) the morphismassociated to OY (1). Given a generic linear subspace L ⊂ V ∗, consider the linearsections XL := X ×P(V ∗) P(L⊥) and YL := Y ×P(V ) P(L). The next result, whoseproof makes essential use of the above Theorem 3.2, is obtained by concatenating[52, §5.3-5.4] with [56, Thm. 1.4] and [57, Thm. 1.1].

Theorem 3.7 (HPD-invariance6). Let X and Y be as above. Assume that XL

and YL are smooth, that dim(XL) = dim(X)− dim(L), that dim(YL) = dim(Y )−dim(L⊥), and that one of the following conjectures holds

(3.8) C+nc(A0,dg) Dnc(A0,dg) Vnc(A0,dg) Snc(A0,dg) ,

where A0,dg stands for the dg enhancement of A0 induced by perfdg(X). Underthese assumptions, we have the corresponding equivalence of conjectures

C+(XL)⇔ C+(YL) D(XL)⇔ D(YL) V (XL)⇔ V (YL) S(XL)⇔ S(YL) .

Remark 3.9. Conjectures (3.8) hold, in particular, whenever the triangulated cat-egory A0 admits a full exceptional collection. Furthermore, Theorem 3.7 holdsmore generally when Y (or X) is singular. In this case, we need to replace Y by anoncommutative resolution of singularities perf(Y ;F) in the sense of [37, §2.4].

Roughly speaking, Theorem 3.7 shows that the original conjectures C+, D, V, Sare invariant under homological projective duality. As a consequence of this invari-ance, we obtain the following practical result:

Corollary 3.10. Let XL and YL be smooth linear sections as in Theorem 3.7.(i) When dim(YL) ≤ 2, the conjecture C+(XL) holds;

(ii) When dim(YL) ≤ 4, the conjecture D(XL) holds;(iii) When dim(YL) ≤ 2, the conjecture V (XL) holds;(iv) When dim(YL) ≤ 1, the conjecture S(XL) holds.

In Examples 3.11, 3.13, and 3.15 below, we illustrate the strength of Corollary3.10; many more examples can be found in [56, 57].

Example 3.11 (Spinor duality). Let W be a 10-dimensional k-vector space andq ∈ S2W ∗ a nondegenerate quadratic form. The associated isotropic Grassman-nian of 5-dimensional subspaces in W has two (isomorphic) connected components

6Consult Theorem 6.3 below for another HPD-invariance type result.

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 11

X := Sp+(5, 10) ⊂ P(∧5W ) and Y := Sp−(5, 10) ⊂ P(∧5W ∗) called the Spinorvarieties. As explained by Kuznetsov in [39, §6.2], the category perf(Sp+(5, 10))admits a Lefschetz decomposition 〈A0,A1(1), . . . ,A7(7)〉 and A0 a full exceptionalcollection. Moreover, the spinor varieties Sp+(5, 10) and Sp−(5, 10) are HP-dualto each other. Given a generic linear subspace L ⊂ ∧5W ∗, consider the associatedsmooth linear sections Sp+(5, 10)L and Sp−(5, 10)L. Making use of Corollary 3.10and of the equalities dim(Sp+(5, 10)) = 10, dim(Sp+(5, 10)L) = 10 − dim(L), anddim(Sp−(5, 10)L) = dim(L)− 6, we obtain the following result:

Corollary 3.12. Let Sp+(5, 10)L be a smooth linear section as above.(i) The conjecture C+(Sp+(5, 10)L) holds when dim(L) ≤ 8;(ii) The conjecture D(Sp+(5, 10)L) holds;

(iii) The conjecture V (Sp+(5, 10)L) holds when dim(L) ≤ 8;(iv) The conjecture S(Sp+(5, 10)L) holds when dim(L) ≤ 7.

Example 3.13 (Grassmannian-Pfaffian duality). Let W be a k-vector space of di-mension 6, resp. 7, and X the Grassmannian Gr(2, 6), resp. Gr(2, 7), equippedwith the Plucker embedding Gr(2, 6) → P(∧2W ), resp. Gr(2, 7) → P(∧2W ). Asexplained by Kuznetsov in [40, §10], resp. in [40, §11], the category perf(Gr(2, 6)),resp. perf(Gr(2, 7)), admits a Lefschetz decomposition 〈A0,A1(1), . . . ,A5(5)〉, resp.〈A0,A1(1), . . . ,A6(6)〉, and A0 a full exceptional collection. Furthermore, the HP-dual Y of Gr(2, 6), resp. Gr(2, 7), is given by perf(Pf(4,W ∗);F), where Pf(4,W ∗) ⊂P(∧2W ∗) is the (singular) Pfaffian variety and F a certain coherent sheaf of al-gebras on Pf(4,W ∗). The singular locus of Pf(4,W ∗) is 8-dimensional, resp. 10-dimensional, and F is Morita equivalent to the structure sheaf on the smooth locus.Therefore, given a generic linear subspace L ⊂ ∧2W ∗ of dimension ≤ 6, resp. ≤ 10,we can consider the associated smooth linear sections Gr(2, 6)L and Pf(4,W ∗)L,resp. Gr(2, 7)L and Pf(4,W ∗)L. Making use of Corollary 3.10 and of the equalitiesdim(Gr(2, 6)) = 8, dim(Gr(2, 6)L) = 8−dim(L), and dim(Pf(4,W ∗)L) = dim(L)−2, resp. dim(Gr(2, 7)) = 10, dim(Gr(2, 7)L) = 10−dim(L), and dim(Pf(4,W ∗)L) =dim(L)− 4, we obtain the following result:

Corollary 3.14. Let Gr(2, 6)L and Gr(2, 7)L be smooth linear sections as above.(i) The conjecture C+(Gr(2, 6)L) holds when dim(L) ≤ 4;(i’) The conjecture C+(Gr(2, 7)L) holds when dim(L) ≤ 6;(ii) The conjecture D(Gr(2, 6)L) holds when dim(L) ≤ 6;

(ii’) The conjecture D(Gr(2, 7)L) holds;(iii) The conjecture V (Gr(2, 6)L) holds when dim(L) ≤ 4;(iii’) The conjecture V (Gr(2, 7)L) holds when dim(L) ≤ 6;(iv) The conjecture S(Gr(2, 6)L) holds when dim(L) ≤ 3;(iv’) The conjecture S(Gr(2, 7)L) holds when dim(L) ≤ 5.

Example 3.15 (Determinantal duality). Let U and V be two k-vector spaces of di-mensions m and n, respectively, with m ≤ n, W the tensor product U ⊗ V , and0 < r < m an integer. Consider the determinantal variety Zrm,n ⊂ P(W ), resp.Wrm,n ⊂ P(W ∗), defined as the locus of those matrices V → U∗, resp. V ∗ → U ,

with rank at most r, resp. with corank at least r. For example, Z1m,n are the

classical Segre varieties. As explained by Bernardara, Bolognesi, and Faenzi in[4, §3], Zrm,n and Wr

m,n admit (Springer) resolutions of singularities X := Xrm,n

and Y := Y rm,n, respectively. Moreover, the category perf(Xrm,n) admits a Lef-

schetz decomposition 〈A0,A1(1), . . . ,Anr−1〉 and A0 a full exceptional collection.

12 GONCALO TABUADA

Furthermore, the resolutions Xrm,n and Y rm,n are HP-dual to each other. Given

a generic linear subspace L ⊂ W ∗, consider the associated smooth linear sec-tions (Xr

m,n)L and (Y rm,n)L. Making use of Corollary 3.10 and of the equalitiesdim(Xr

m,n) = r(n + m − r) − 1, dim((Xrm,n)L) = r(n + m − r) − 1 − dim(L), and

dim((Y rm,n)L) = r(m− n− r)− 1 + dim(L), we obtain the following result:

Corollary 3.16. Let (Xrm,n)L be a smooth linear section as above.

(i) The conjecture C+((Xrm,n)L) holds when dim(L) ≤ 3− r(m− n− r);

(ii) The conjecture D((Xrm,n)L) holds when dim(L) ≤ 5− r(m− n− r);

(iii) The conjecture V ((Xrm,n)L) holds when dim(L) ≤ 3− r(m− n− r);

(iv) The conjecture S((Xrm,n)L) holds when dim(L) ≤ 2− r(m− n− r).

Note that the dimension of the smooth linear section (Xrm,n)L, i.e. the integer

r(n + m − r) − 1 − dim(L), can be arbitrary high. Consequently, Corollary 3.16provides infinitely many examples of smooth projective k-schemes, of arbitrary highdimension, which satisfy the conjectures C+, D, V , and S.

Remark 3.17 (Kimura-finiteness7). The Kimura-finiteness conjecture, denoted byK(X), asserts that the Chow motive h(X)Q is Kimura-finite in the sense of [29, §7];see [52, §3.0.11]. This conjecture is stronger than conjecture S(X). However, asproved in [57, Thm. 1.10], all the above Corollaries 3.12, 3.14, and 3.16, still holdwith conjecture S replaced by conjecture K.

3.1.2. Smooth proper algebraic stacks. Theorem 3.2 allows us to extend the originalconjectures C+, D, V, S to smooth proper algebraic stacks X by setting ?(X ) :=?nc(perfdg(X )), where ? ∈ {C+, D, V, S}. In some low-dimensional cases, we areable to prove these conjectures. Here are two illustrative examples:

Example 3.18 (“Low-dimensional” orbifolds). Let G be a finite group and X asmooth projective k-scheme equipped with a G-action. Similarly to §1.2, let uswrite [X/G] for the associated (global) orbifold.

Theorem 3.19 ([62, Thm. 9.2]). Let [X/G] be a (global) orbifold as above.(i) The conjecture C+([X/G]) holds when dim(X) ≤ 2 or when X is an abelian

variety and G acts by group homomorphisms;(ii) The conjecture D([X/G]) holds when dim(L) ≤ 4;(iii) The conjecture V ([X/G]) holds when dim(L) ≤ 2;(iv) The conjecture S([X/G]) holds when dim(L) ≤ 1.

Example 3.20 (Intersection of bilinear divisors). Let W be a k-vector space ofdimension d, and X the smooth proper Deligne-Mumford stack (P(W )×P(W ))/µ2

equipped with the map X → P(S2W ), ([w1], [w2]) 7→ [w1 ⊗ w2 + w2 ⊗ w1]. Givena generic linear subspace L ⊂ S2W ∗, the linear section XL corresponds to theintersection of the dim(L) bilinear divisors in X parametrized by L.

Theorem 3.21 ([56, Thm. 1.13]). Let XL be a linear section as above with d odd.(i) The conjecture C+(XL) holds when dim(L) ≤ 3;

(ii) The conjecture D(XL) holds when dim(L) ≤ 5;(iii) The conjecture V (XL) holds when dim(L) ≤ 3;(iv) The conjecture S(XL) holds when dim(L) ≤ 2.

7Consult Theorem 5.13 below for a Kimura-finiteness result in the setting of mixed motives.

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 13

4. Localizing invariants

Recall from [52, §8.1] the notion of a short exact sequence of dg categories in thesense of Drinfeld/Keller. In §4.1 below, we describe a key structural property ofthese short exact sequences, and explain its implication to secondary K-theory.

Recall from [52, §8.5.1] the construction of the universal localizing A1-invariantof dg categories U: dgcat(k)→ NMot(k); in loc. cit. we used the more explicit no-

tation UA1

loc : dgcat(k)→ NMotA1

loc(k). In [52, §8.5.3], we described the behavior of Uwith respect to dg orbit categories and dg cluster categories. In §4.2-4.3 below, wedescribe the behavior of U with respect to open/closed scheme decompositions andto corner skew Laurent polynomial algebras. As explained in [52, Thm. 8.25], allthe results of §4.2-4.3 are motivic in the sense that they hold similarly for every lo-calizing A1-invariant such as mod-lν algebraic K-theory (when 1/l ∈ k), homotopyK-theory, etale K-theory, periodic cyclic homology8 (when char(k) = 0), etc.

In order to simplify the exposition, given a k-scheme X (or more generally analgebraic stack X ), we will write U(X) instead of U(perfdg(X)).

4.1. Short exact sequences. Recall from [52, §8.4] the notion of a split shortexact sequence of dg categories 0 → A → B → C → 0. Up to Morita equivalence,this data is equivalent to inclusions of dg categories A, C ⊆ B yielding a semi-orthogonal decomposition of triangulated categories H0(B) = 〈H0(A),H0(C)〉 inthe sense of Bondal-Orlov [10]; by definition, the dg category H0(A) has the sameobjects as A and morphisms H0(A)(x, y) := H0A(x, y).

Theorem 4.1 ([51, Thm. 4.4]). Let 0→ A→ B → C → 0 be a short exact sequenceof dg categories in the sense of Drinfeld/Keller. If A is smooth and proper and Bis proper, then the short exact sequence is split.

Morally speaking, Theorem 4.1 shows that the smooth proper dg categories be-have as “injective” objects. In the setting of triangulated categories, this conceptualidea goes back to the pioneering work of Bondal and Kapranov [8]. In §4.1.1 below,we explain the implications of Theorem 4.1 to secondary K-theory.

4.1.1. Secondary K-theory. Two decades ago, Bondal, Larsen, and Lunts intro-duced in [9] the Grothendieck ring of smooth proper dg categories PT (k). Thisring is defined by generators and relations. The generators are the Morita equiva-lence classes of smooth proper dg categories9 and the relations [B] = [A] + [C] arisefrom semi-orthogonal decompositions H0(B) = 〈H0(A),H0(C)〉. The multiplicationlaw is induced by the tensor product of dg categories. One decade ago, Toen in-troduced in [64] a “categorified” version of the Grothendieck ring named secondary

Grothendieck ring K(2)0 (k). By definition, K

(2)0 (k) is the quotient of the free abelian

group on the Morita equivalence classes of smooth proper dg categories by the re-lations [B] = [A] + [C] arising from short exact sequences 0 → A → B → C → 0.The multiplication law is also induced by the tensor product of dg categories.

The above Theorem 4.1 directly leads to the following result:

8Periodic cyclic homology is not a localizing A1-invariant in the sense of [52, §8.5] because itdoes not preserves filtered (homotopy) colimits. Nevertheless, as illustrated in §4.2.2 below, allthe results of §4.2-4.3 similarly hold for periodic cyclic homology.

9Bondal, Larsen, and Lunts originally worked with (pretriangulated) dg categories. In thisgenerality, the classical Eilenberg’s swindle argument implies that the Grothendieck ring is trivial.

14 GONCALO TABUADA

Corollary 4.2. The rings PT (k) and K(2)0 (k) are isomorphic.

By construction, the universal additive invariant U (see §1) sends semi-orthogonaldecompositions to direct sums. Therefore, it gives rise to a ring homomorphismPT (k)→ K0(NChow(k)). Making use of Corollary 2.5, we then obtain the result:

Corollary 4.3. The following map is injective

Br(k) −→ PT (k) ' K(2)0 (k) [A] 7→ [A] .(4.4)

Intuitively speaking, (4.4) may be understood as the “categorification” of thecanonical map from the Picard group Pic(k) to the Grothendieck ring K0(k). Incontrast with Pic(k)→ K0(k), (4.4) does not seem to admit a “determinant” mapin the converse direction. Nevertheless, Corollary 4.3 shows that (4.4) is injective.

Remark 4.5 (Generalizations). The results of §4.1 hold more generally in the casewhere k is a commutative ring, and Remark 2.6 also applies to Corollary 4.3.

4.2. Gysin triangle. Let X be a smooth k-scheme, i : Z ↪→ X a smooth closedk-subscheme, and j : V ↪→ X the open complement of Z. The next result describesthe behavior of U with respect to the open V /closed Z decomposition of X:

Theorem 4.6 ([61, Thm. 1.9]). We have an induced distinguished “Gysin” triangle

(4.7) U(Z)U(i∗)−→ U(X)

U(j∗)−→ U(V )∂−→ ΣU(Z) ,

where i∗, resp. j∗, stands for the push-forward, resp. pull-back, dg functor.

Remark 4.8 (Generalizations). As explained in [61, §7], Theorem 4.6 holds not onlyfor smooth schemes but also for smooth algebraic spaces in the sense of Artin.

Consult Remark 5.5 below, resp. 5.9, for the relation between (4.7) and the mo-tivic Gysin triangles originally constructed by Morel-Voevodsky, resp. Voevodsky.

Roughly speaking, Theorem 4.6 shows that the difference between the local-izing A1-invariants of X and U is completely determined by the smooth closedk-subscheme Z. Let us denote by perfdg(X)Z the dg category of those perfect com-plexes of OX -modules that are supported on Z. Thanks to the work of Thomason-Trobaugh [63, §5], the bulk of Theorem 4.6 is the fact that the induced morphismU(i∗) : U(Z)→ U(perfdg(X)Z) is invertible. Among other ingredients, the proof ofthis latter fact is based on a description of the dg category perfdg(X)Z in terms ofa formal dg k-algebra; consult [61, §6] for details.

4.2.1. Quillen’s localization theorem. HomotopyK-theory is a localizing A1-invariantwhich agrees with Quillen’s algebraicK-theory when restricted to smooth k-schemes.Therefore, Theorem 4.6 leads to the localization theorem in algebraic K-theory

(4.9) K(Z)K(i∗)−→ K(X)

K(j∗)−→ K(U)∂−→ ΣK(Z)

originally established by Quillen in [49, Chapter 7 §3]. Quillen’s proof is based ondevissage and on the equivalence between K-theory and G-theory. As mentionedabove, the proof of Theorem 4.6, and consequently of (4.9), is quite different.

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 15

4.2.2. Six-term exact sequence in de Rham cohomology. Periodic cyclic homology isa localizing A1-invariant (when char(k) = 0). Moreover, thanks to the Hochschild-Kostant-Rosenberg theorem, the Z/2-graded k-vector space HP±(X) identifieswith (

⊕n evenH

ndR(X),

⊕n oddH

ndR(X)), where H∗dR(−) stands for de Rham co-

homology. Furthermore, the maps i : Z ↪→ X and j : V ↪→ X give rise to homomor-phisms Hn

dR(i∗) : HndR(Z) → Hn+2c

dR (X) and HndR(j∗) : Hn

dR(X) → HndR(V ) where

c stands for the codimension of i. Therefore, Theorem 4.6 leads to the followingsix-term exact sequence in de Rham cohomology:⊕

n evenHndR(Z)

⊕nH

ndR(i∗) //⊕

n evenHndR(X)

⊕nH

ndR(j∗) //⊕

n evenHndR(V )

∂��⊕

n oddHndR(V )

∂

OO

⊕n oddH

ndR(X)⊕

nHndR(j∗)

oo ⊕n oddH

ndR(Z) .⊕

nHndR(i∗)

oo

This exact sequence is the “2-periodization” of the Gysin long exact sequence onde Rham cohomology originally constructed by Hartshorne in [19, Chapter II §3].

4.2.3. Reduction to smooth projective schemes. As a byproduct of Theorem 4.6, thestudy of the localizing A1-invariants of smooth k-schemes can be reduced to thestudy of the localizing A1-invariants of smooth projective k-schemes:

Theorem 4.10 ([61, Thm. 2.1]). Let X a smooth k-scheme.(i) When k is of characteristic 0, the object U(X) belongs to the smallest trian-

gulated subcategory of NMot(k) containing the objects U(Y ), with Y a smoothprojective k-scheme;

(ii) When k is a perfect field of characteristic p > 0, the object U(X)1/p belongsto the smallest thick triangulated subcategory of NMot(k)1/p containing theobjects U(Y )1/p, with Y a smooth projective k-scheme.

In addition to Theorem 4.6, the proof of item (i), resp. item (ii), makes use ofresolution of singularities, resp. of Gabber’s refined version of de Jong’s theory ofalterations; consult [61, §8] for details.

Remark 4.11 (Dualizable objects). Given a smooth projective k-scheme Y , the as-sociated dg category perfdg(Y ) is smooth and proper; see [52, Ex. 1.42]. Therefore,

since the universal localizing A1-invariant U is symmetric monoidal, it follows from[52, Thm. 1.43] that U(Y ) is a dualizable object of the symmetric monoidal cate-gory NMot(k). Given a smooth k-scheme X, the associated dg category perfdg(X)is smooth but not necessarily proper. Nevertheless, thanks to Theorem 4.10, weconclude that U(X), resp. U(X)1/n, is still a dualizable object of the symmetricmonoidal category NMot(k), resp. NMot(k)1/p.

4.3. Corner skew Laurent polynomial algebras. Let A be a unital k-algebra,e an idempotent of A, and φ : A

∼→ eAe a “corner” isomorphism. The associatedcorner skew Laurent polynomial algebra A[t+, t−;φ] is defined as follows: the el-ements are formal expressions tm−a−m + · · · + t−a−1 + a0 + a1t+ · · · + ant

n+ with

a−i ∈ φi(1)A and ai ∈ Aφi(1) for every i ≥ 0; the addition is defined component-wise; the multiplication is determined by the distributive law and by the relationst−t+ = 1, t+t− = e, at− = t−φ(a) for every a ∈ A, and t+a = φ(a)t+ for everya ∈ A. Note that A[t+, t−;φ] admits a canonical Z-grading with deg(t±) = ±1. Asproved in [2, Lem. 2.4], the corner skew Laurent polynomial algebras can be charac-terized as those Z-graded algebras C =

⊕n∈Z Cn containing elements t+ ∈ C1 and

16 GONCALO TABUADA

t− ∈ C−1 such that t−t+ = 1. Concretely, we have C = A[t+, t−;φ] with A := C0,e := t+t−, and φ : C0 → t+t−C0t+t− given by c0 7→ t+c0t−.

Example 4.12 (Skew Laurent polynomial algebras). When e = 1, A[t+, t−;φ] re-duces to the classical skew Laurent polynomial algebra A oφ Z. In the particularcase where φ is the identity, Aoφ Z reduces furthermore to A[t, t−1].

Example 4.13 (Leavitt algebras). Following [41], the Leavitt algebra Ln, n ≥ 0, isthe k-algebra generated by elements x0, . . . , xn, y0, . . . , yn subject to the relationsyixj = δij and

∑ni=0 xiyi = 1. Note the canonical Z-grading, with deg(xi) = 1 and

deg(yi) = −1, makes Ln into a corner skew Laurent polynomial algebra. Note alsothat L0 ' k[t, t−1]. In the remaining cases n ≥ 1, Ln is the universal example of a

k-algebra of module type (1, n+ 1), i.e. Ln ' L⊕(n+1)n as right Ln-modules.

Example 4.14 (Leavitt path algebras). Let Q = (Q0, Q1, s, r) be a finite quiver;Q0 and Q1 stand for the sets of vertices and arrows, respectively, and s and rfor the source and target maps, respectively. We assume that Q has no sources,i.e. vertices i ∈ Q0 such that {α | r(α) = i} = ∅. Consider the double quiverQ = (Q0, Q1 ∪ Q∗1, s, r) obtained from Q by adding an arrow α∗, in the conversedirection, for each arrow α ∈ Q1. The Leavitt path algebra LQ of Q is the quotient

of the quiver algebra kQ (which is generated by elements α ∈ Q1 ∪ Q∗1 and eiwith i ∈ Q0) by the Cuntz-Krieger’s relations: α∗β = δαβer(α) for every α, β ∈ Q1

and∑{α∈Q1|s(α)=i} αα

∗ = ei for every non-sink i ∈ Q0. Note that LQ admits a

canonical Z-grading with deg(α) = 1 and deg(α∗) = −1. For every vertex i ∈ Q0

choose an arrow αi such that r(αi) = i and consider the associated elements t+ :=∑i∈Q0

αi and t− := t∗+. Since deg(t±) = ±1 and t−t+ = 1, LQ is an example ofa corner skew Laurent polynomial algebra. In the particular case where Q is thequiver with one vertex and n+ 1 arrows, LQ is isomorphic to Ln.

Theorem 4.15 ([58, Thm. 3.1]). We have an induced distinguished triangle

U(A)id−U(φA)−→ U(A) −→ U(A[t+, t−;φ])

∂−→ ΣU(A) ,

where φA stands for the A-A-bimodule associated to φ.

Intuitively speaking, Theorem 4.15 shows that the corner skew Laurent polyno-mial algebra A[t+, t−;φ] can be thought of as a model for the orbits of the N-actionon U(A) induced by the endomorphism U(φA). Among other ingredients, its proofmakes use of a description of A[t+, t−;φ] in terms of a dg orbit category in the senseof Keller [28]; consult [58, §6] for details.

4.3.1. Leavitt path algebras. Let Q = (Q0, Q1, s, r) be a quiver as in Example 4.14,with v vertices and v′ sinks. Assume that the set Q0 is ordered with the first v′

elements corresponding to the sinks. Let I ′Q be the incidence matrix of Q, IQ the

matrix obtained from I ′Q by removing the first v′ rows (which are zero), and ItQ the

transpose of IQ. Under these notations, the above Theorem 4.15 (with corner skewLaurent polynomial algebra LQ) admits the following refinement:

Theorem 4.16 ([58, Thm. 3.7]). We have an induced distinguished triangle:

(4.17) ⊕v−v′

i=1 U(k)( 0id)−I

tQ−→ ⊕vi=1U(k) −→ U(LQ)

∂−→ ⊕v−v′

i=1 ΣU(k) .

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 17

Roughly speaking, Theorem 4.16 shows that all the information concerning lo-calizing A1-homotopy invariants of Leavitt path algebras LQ is encoded in theincidence matrix of the quiver Q. In particular, Theorem 4.16 directly leads to thefollowing explicit model for the mod-n Moore construction10:

Example 4.18 (mod-n Moore construction). Let Q be the quiver with one vertexand n+ 1 arrows. In this particular case, (4.17) reduces to

U(k)n·id−→ U(k) −→ U(Ln)

∂−→ ΣU(k) .

This shows that the Leavitt algebra Ln, n ≥ 2, is a model for the mod-n Mooreobject of U(k). Therefore, since the universal localizing A1-invariant U is symmetricmonoidal, given a small dg category A, we conclude that the tensor product A⊗Lnis a model for the mod-n Moore object of U(A).

5. Noncommutative mixed motives

In this section we assume that k is perfect. Kontsevich introduced in [33, 34, 35]a certain rigid symmetric monoidal triangulated category of noncommutative mixedmotives NMix(k). As explained in [52, §9.1.1], this category can be described as thesmallest thick triangulated subcategory of NMot(k) (see §4) containing the objectsU(A), with A a smooth proper dg category. In the same vein, let NMix(k)⊕ be thesmallest triangulated subcategory of NMot(k) which contains NMix(k) and is stableunder arbitrary direct sums. In §5.1 below, we discuss the computation of the Pi-card group of the thick triangulated subcategory of NMix(k) generated by the non-commutative mixed motives of central simple k-algebras. In §5.2-5.4, we describethe relation between NMix(k) and Morel-Voevodsky’s stable A1-homotopy categorySH(k), Voevodsky’s triangulated category of geometric mixed motives DMgm(k),and Levine’s triangulated category of mixed motives DM(k), respectively. Finally,in §5.5 we discuss Kimura-finiteness of quadric fibrations over smooth curves.

5.1. Picard group. The computation of the Picard group of the category of non-commutative mixed motives is a major challenge which seems completely out ofreach at the present time. However, as we explain below, this major challenge canbe met if we restrict ourselves to central simple k-algebras.

Let NMixcsa(k) be the thick triangulated subcategory of NMix(k) generated bythe noncommutative mixed motives U(A) of central simple k-algebras A. Similarlyto §2.1, we have [A] = [B]⇔ U(A) ' U(B) for any two central simple k-algebras Aand B. Moreover, as explained in [52, Thm. 8.28], we have non-trivial Ext-groups

(5.1) HomNMix(k)(U(A),U(B)[−n]) ' Kn(Aop ⊗B) n ∈ Z .

This shows that NMixcsa(k) contains information not only about the Brauer groupBr(k) but also about all the higher algebraic K-theory of central simple k-algebras.

Theorem 5.2 ([11, Thm. 2.22]). We have the following isomorphism

Br(k)× Z ∼−→ Pic(NMixcsa(k)) ([A], n) 7→ ΣnU(A) .

10Explicit models for the suspension construction, namely the Waldhausen’s S•-constructionand the Calkin algebra, are described in [52, §8.3.2 and §8.4.4].

18 GONCALO TABUADA

Morally speaking, Theorem 5.2 shows that although the category NMixcsa(k)contains information about all the higher algebraic K-theory of central simple k-algebras, none of the noncommutative mixed motives which are built using thenon-trivial Ext-groups (5.1) is ⊗-invertible. Among other ingredients, its proofmakes use of a general theory which enables the computation of the Picard groupof a symmetric monoidal triangulated category equipped with a weight structurein terms of the Picard group of the associated heart; consult [11, §10] for details.

5.2. Morel-Voevodsky’s motivic category. Morel and Voevodsky introducedin [46, 67] the stable A1-homotopy category of (P1,∞)-spectra SH(k). By construc-tion, we have a symmetric monoidal functor Σ∞(−+) : Sm(k)→ SH(k) defined onsmooth k-schemes. Let KGL ∈ SH(k) be the ring (P1,∞)-spectrum representinghomotopy K-theory and Mod(KGL) the homotopy category of KGL-modules.

Theorem 5.3. (i) When char(k) = 0, there exists a fully-faithful, symmetricmonoidal, triangulated functor Ψ making the following diagram commute

(5.4) Sm(k)

Σ∞(−+)��

X 7→perfdg(X)//

**

dgcat(k)

U��

SH(k)

−∧KGL��

NMix(k)

(−)∨

��

// NMot(k)

Hom(−,U(k))��

Mod(KGL)Ψ

// NMix(k)⊕ // NMot(k) ,

where Hom(−,−) stands for the internal-Hom of the closed symmetric monoidalcategory NMot(k) and (−)∨ for the (contravariant) duality functor;

(ii) When char(k) = p > 0, there exists a Z[1/p]-linear, fully-faithful, symmetricmonoidal, triangulated functor Ψ1/p making the following diagram commute:

Sm(k)

Σ∞(−+)1/p��

X 7→perfdg(X)//

**

dgcat(k)

U(−)1/p��

SH(k)1/p

−∧KGL1/p

��

NMix(k)1/p

(−)∨

��

// NMot(k)1/p

Hom(−,U(k)1/p)��

Mod(KGL1/p) Ψ1/p

// NMix(k)⊕1/p// NMot(k)1/p .

Intuitively speaking, Theorem 5.3 shows that as soon as we pass to KGL-modules, the commutative world embeds fully-faithfully into the noncommutativeworld. Consult [52, §9.4], and the references therein, for the construction of thetwo outer commutative diagrams. The inner commutative diagrams follow fromthe combination of Theorem 4.10 with Remark 4.11 (see [61, Thm. 3.1]).

Remark 5.5 (Morel-Voevodsky’s motivic Gysin triangle). Let X be a smooth k-scheme, i : Z ↪→ X a smooth closed k-subscheme with normal vector bundle N ,and j : V ↪→ X the open complement of Z. Making use of homotopy purity, Moreland Voevodsky constructed in [46, §3.2][67, §4] a motivic Gysin triangle

(5.6) Σ∞(V+)Σ∞(j+)−→ Σ∞(X+) −→ Σ∞(Th(N))

∂−→ Σ(Σ∞(V+))

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 19

in SH(k), where Th(N) stands for the Thom space of N . Since homotopy K-theoryis an orientable and periodic cohomology theory, Σ∞(Th(N))∧KGL is isomorphicto Σ∞(Z+) ∧ KGL. Using the above commutative diagram (5.4), we observe thatthe image of (5.6) under the composed functor Ψ◦ (−∧KGL): SH(k)→ NMix(k)⊕

agrees with the dual of the Gysin triangle (4.7). In other words, the Gysin triangle(4.7) is the dual of the “KGL-linearization” of (5.6).

5.3. Voevodsky’s motivic category. Voevodsky introduced in [66, §2] the trian-gulated category of geometric mixed motives DMgm(k). By construction, this cat-egory comes equipped with a symmetric monoidal functor M : Sm(k)→ DMgm(k)and is the natural setting for the study of algebraic cycle (co)homology theoriessuch as higher Chow groups, Suslin homology, motivic cohomology, etc.

Theorem 5.7. There exists a Q-linear, fully-faithful, symmetric monoidal functorΦQ making the following diagram commute:

(5.8) Sm(k)

M(−)Q��

X 7→perfdg(X)//

++

dgcat(k)

U(−)Q��

DMgm(k)Q

��

NMix(k)Q

(−)∨

��

// NMot(k)Q

Hom(−,U(k)Q)��

DMgm(k)Q/−⊗Q(1)[2] ΦQ

// NMix(k)Q // NMot(k)Q .

Intuitively speaking, Theorem 5.7 shows that as soon as we “⊗-trivialize” theTate motive Q(1)[2], the commutative world embedds fully-faithfully into the non-commutative world. Consult [52, §9.5], and the references therein, for the construc-tion of the outer commutative diagram. The inner commutative diagram followsfrom the combination of Theorem 4.10 with Remark 4.11 (see [61, Thm. 3.7]).

Remark 5.9 (Voevodsky’s motivic Gysin triangle). Let X be a smooth k-scheme,i : Z ↪→ X a smooth closed k-subscheme of codimension c, and j : V ↪→ X theopen complement of Z. Making use of deformation to the normal cone, Voevodskyconstructed in [66, §2] a motivic Gysin triangle

(5.10) M(V )QM(j)Q−→ M(X)Q −→M(Z)Q(c)[2c]

∂−→ ΣM(V )Q

in the category DMgm(k)Q. Using the above commutative diagram (5.8), we observethat the image of (5.10) under the (composed) functor ΦQ : DMgm(k)Q → NMix(k)Qagrees with the dual of the rationalized Gysin triangle (4.7). In other words, therationalized Gysin triangle (4.7) is the dual of the “Tate ⊗-trivialization” of (5.10).

Let DMetgm(k) be the etale variant of DMgm(k) constructed by Voevodsky in [66,

§3.3]. As proved in loc. cit., the category DMgm(k)Q is equivalent to DMetgm(k)Q.

Consequently, Theorem 5.7 leads to the following result (see [61, Thm. 3.13]):

Corollary 5.11 (Etale descent). The presheaf of noncommutative mixed motives

Sm(k)op −→ NMot(k)Q X 7→ U(X)Q

satisfies etale descent, i.e. for every etale cover V = {Vi → X}i∈I of X, we have aninduced isomorphism U(X)Q ' holimn≥0U(CnV)Q, where C•V stands for the Cechsimplicial k-scheme associated to the cover V.

20 GONCALO TABUADA

5.4. Levine’s motivic category. Levine introduced in [43, Part I] a certain tri-angulated category of mixed motives DM(k) and a (contravariant) symmetricmonoidal functor h : Sm(k) → DM(k). As proved in [20, Thm. 4.2], the as-signment h(X)Q(n) 7→ Hom(M(X),Q(n)) gives rise to an equivalence of cate-gories DM(k)Q → DMgm(k)Q whose precomposition with h(−)Q agrees with X 7→M(X)∨Q. Thanks to Theorem 5.7, there exists then a Q-linear, fully-faithful, sym-metric monoidal functor ΦQ making the following diagram commute:

(5.12) Sm(k)

h(−)Q��

X 7→perfdg(X)// dgcat(k)

U(−)Q

��

DM(k)Q

��DM(k)Q/−⊗Q(1)[2] ΦQ

// NMix(k)Q // NMot(k)Q .

Note that in contrast with Theorems 5.3 and 5.7, the above commutative diagram(5.12) does not use any kind of duality functor.

5.5. Kimura-finiteness. An important open problem11 is the classification of allthe Kimura-finite geometric mixed motives and the computation of the correspond-ing Kimura-dimensions. On the negative side, O’Sullivan constructed a certainsmooth surface S whose mixed motive M(S)Q is not Kimura-finite; consult [45,§5.1] for details. On the positive side, Guletskii [18] and Mazza [45] proved, inde-pendently, that the mixed motive M(C)Q of every smooth curve C is Kimura-finite.

Theorem 5.13 ([60, Thm. 1.1]). Let q : Q → C be a flat quadric fibration ofrelative dimension d−2 over a smooth curve C. Assume that Q is smooth and thatq has only simple degenerations, i.e. that all the fibers of q have corank ≤ 1.(i) When d is even, the mixed motive M(Q)Q is Kimura-finite. Moreover, we have

kim(M(Q)Q) = kim(M(C)Q) + (d− 2)kim(M(C)Q) ,

where D ↪→ C stands for the finite set of critical values of q and C for thediscriminant double cover of C (ramified over D).

(ii) When d is odd, k is algebraically closed, and 1/2 ∈ k, the mixed motive M(Q)Qis Kimura-finite. Moreover, we have the following equality:

kim(M(Q)Q) = #D + (d− 1)kim(M(C)Q) .

Intuitively speaking, Theorem 5.13 bootstraps Kimura-finiteness from smoothcurves to families of quadrics over smooth curves. Among other ingredients, itsproof makes use in an essential way of the above commutative diagram (5.8).

6. Periods of differential graded categories

In this section we assume that k is equipped with an embedding into the complexnumbers C. Consider the Z-graded C-algebra of Laurent polynomials C[t, t−1] witht of degree 1. Let Vect(k,Q) be the category of triples (V,W, ω), where V is a finitedimensional k-vector space, W a finite dimensional Q-vector space, and ω an isomor-phism V ⊗kC

∼→W⊗QC. Given such a triple (V,W, ω), let us write P(V,W, ω) ⊆ Cfor the subset of entries of the matrix representations of ω (with respect to basis

11Among other consequences, Kimura-finiteness implies rationality of the motivic zeta function.

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 21

of V and W ). In the same vein, given an object {(Vn,Wn, ωn)}n∈Z of the categoryVectZ(k,Q), let us write P({(Vn,Wn, ωn)}n∈Z) for the Z-graded k-subalgebra ofC[t, t−1] generated in degree n by the elements of the set P (Wn,Wn, ωn). In thecase of a k-scheme X, P(X) := P(H∗dRB(X)) is called the (Z-graded) algebra ofperiods of X, where H∗dRB(−) stands for de Rham-Betti cohomology. This algebra,originally introduced by Grothendieck in the sixties, plays a key role in the study oftranscendental numbers; consult, for example, the work of Kontsevich-Zagier [36].

Consider the Z/2-graded C-algebra C2πiZ/2 := C[t, t−1]/〈1− (2πi)t2〉 and the asso-

ciated quotient homomorphism φ : C[t, t−1] � C2πiZ/2. The following result extends

the aforementioned theory of periods to the noncommutative world:

Theorem 6.1 ([54, Thm. 4.1]). There exists an assignment A 7→ Pnc(A), withPnc(A) a Z/2-graded k-subalgebra of C2πi

Z/2, satisfying the following properties:

(i) If A = perfdg(X), with X a k-scheme, then Pnc(A) ' φ(P(X));(iii) If A is Morita equivalent to B, then Pnc(A) ' Pnc(B);(iii) If A, C ⊆ B are dg categories yielding, up to Morita equivalence, a semi-

orthogonal decomposition H0(B) = 〈H0(A),H0(C)〉, then Pnc(B) agrees withthe smallest Z/2-graded k-subalgebra of C2πi

Z/2 containing Pnc(A) and Pnc(C).

Corollary 6.2 (Morita invariance). Let X and Y be two k-schemes. If the dgcategories perfdg(X) and perfdg(Y ) are Morita equivalent, then the Z/2-graded k-algebras φ(P(X)) and φ(P(Y )) are isomorphic.

Intuitively speaking, Theorem 6.1 shows that Grothendieck’s theory of periodscan be extended to the noncommutative world as long as we work modulo 2πi.Among other ingredients, its proof makes use of a certain noncommutative versionof de Rham-Betti cohomology theory; consult [54, §1-2] for details12.

Let X and Y be two HP-dual smooth projective k-schemes as in §3.1.1. Recallfrom loc. cit. that the category perf(X) admits, in particular, a Lefschetz de-composition 〈A0,A1(1), . . . ,Ai−1(i− 1)〉. Given a generic linear subspace L ⊂ V ∗,consider the linear sections XL := X ×P(V ∗) P(L⊥) and YL := Y ×P(V ) P(L). Thenext result, proved in [54, Thm. 4.6], relates the algebras of periods of XL and YL.

Theorem 6.3 (HPD-invariance). Let X and Y as above. Assume that XL andYL are smooth, that dim(XL) = dim(X) − dim(L), that dim(YL) = dim(Y ) −dim(L⊥), and that the category A0 admits a full exceptional collection. Under theseassumptions, the Z/2-graded k-algebras φ(P(XL)) and φ(P(YL)) are isomorphic.

Roughly speaking, Theorem 6.3 shows that modulo 2πi the algebra of periodsis invariant under homological projective duality. This result can be applied, inparticular, to the above Examples 3.11, 3.13, and 3.15. A potential application ofTheorem 6.3 is the computation of the transcendental degree of P(XL) using thecorresponding information for P(YL). This is the subject of current research.

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Goncalo Tabuada, Department of Mathematics, MIT, Cambridge, MA 02139, USA

E-mail address: tabuada@math.mit.edu

URL: http://math.mit.edu/~tabuada