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RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES

GONCALO TABUADA

Abstract This survey covers some of the recent developments on noncom-

mutative motives and their applications Among other topics we compute

the additive invariants of relative cellular spaces and orbifolds prove Kontse-vichrsquos semi-simplicity conjecture prove a far-reaching noncommutative gener-

alization of the Weil conjectures prove Grothendieckrsquos standard conjectures

of type C+ and D Voevodskyrsquos nilpotence conjecture and Tatersquos conjec-ture in several new cases embed the (cohomological) Brauer group into sec-

ondary K-theory construct a noncommutative motivic Gysin triangle com-

pute the localizing A1-homotopy invariants of corner skew Laurent polyno-mial algebras and of noncommutative projective schemes relate Kontsevichrsquos

category of noncommutative mixed motives to Morel-Voevodskyrsquos stable A1-homotopy category to Voevodskyrsquos triangulated category of mixed motives

and to Levinersquos triangulated category of mixed motives prove the Schur-

finiteness conjecture for quadric fibrations over low-dimensional bases andfinally extend Grothendieckrsquos theory of periods to the setting of dg categories

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 Society2015] 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 [81]Therefore we suggest the reader to have it at hisher desk while reading thissurvey The monograph [81] is divided into the following chapters

Chapter 1 Differential graded categoriesChapter 2 Additive invariantsChapter 3 Background on pure motivesChapter 4 Noncommutative pure motivesChapter 5 Noncommutative (standard) conjecturesChapter 6 Noncommutative motivic Galois groupsChapter 7 Jacobians of noncommutative Chow motivesChapter 8 Localizing invariantsChapter 9 Noncommutative mixed motivesChapter 10 Noncommutative motivic Hopf dg algebrasAppendix A Grothendieck derivators

Date September 1 2017The 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 thesecond Mid-Atlantic Topology Conference The author was supported by a NSF CAREER Award

1

2 GONCALO TABUADA

In this survey we cover some of the recent developments concerning the Chapters2 4 5 6 8 and 9 These developments are described in Sections 1 2 3 4 5 and6 respectively The final Section 7 entitled ldquoNoncommutative realizations andperiodsrdquo discusses a recent research subject which was not addressed in [81]

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 Kellerrsquos ICM address [45] Inparticular we will freely use the notions of Morita equivalence of dg categories (see[81 sect16]) and smoothproper dg category in the sense of Kontsevich (see [81 sect17])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 [81 Example 127]

1 Additive invariants

Recall from [81 sect23] the construction of the universal additive invariant of dgcategories U dgcat(k) rarr Hmo0(k) In [81 sect24] we described the behavior of Uwith respect to semi-orthogonal decompositions full exceptional collections purelyinseparable field extensions central simple algebras sheaves of Azumaya algebrastwisted flag varieties nilpotent ideals finite-dimensional algebras of finite globaldimension etc In sect11-12 we describe the behavior of U with respect to relativecellular spaces and orbifolds As explained in [81 Thm 29] all the results in sect11-12 are motivic in the sense that they hold similarly for every additive invariant suchas algebraic K-theory mod-n algebraic K-theory Karoubi-Villamayor K-theorynonconnective algebraic K-theory homotopy K-theory etale K-theory Hochschildhomology cyclic homology negative cyclic homology periodic cyclic homologytopological Hochschild homology topological cyclic homology topological periodiccyclic homology etc Consult sect71 for further examples of additive invariants

Notation 11 Given a k-scheme X (or more generally an algebraic stack X ) wewill write U(X) instead of U(perfdg(X))

11 Relative cellular spaces A flat morphism of k-schemes p X rarr Y is calledan affine fibration of relative dimension d if for every point y isin Y there existsa Zariski open neighborhood y isin V such that XV = pminus1(V ) Y times Ad withpV XV rarr Y isomorphic to the projection onto the first factor Following Karpenko[43 Def 61] a smooth projective k-scheme X is called a relative cellular space ifit admits a filtration by closed subschemes

empty = Xminus1 rarr X0 rarr middot middot middot rarr Xi rarr middot middot middot rarr Xnminus1 rarr Xn = X

and affine fibrations pi XiXiminus1 rarr Yi 0 le i le n of relative dimension di with Yia smooth projective k-scheme The smooth k-schemes XiXiminus1 are called the cellsand the smooth projective k-schemes Yi the bases of the cells

Example 12 (Gm-schemes) The celebrated Bialynicki-Birula decomposition [9]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 [43] as well as the isotropichomogeneous spaces considered by Chernousov-Gille-Merkurjev in [16]

RECENT DEVELOPMENTS ON NONCOMMUTATIVE MOTIVES 3

Theorem 13 ([95 Thm 27]) Given a relative cellular space X we have anisomorphism U(X)

oplusni=0 U(Yi)

Theorem 13 shows that the additive invariants of relative cellular spaces Xare completely determined by the basis Yi of the cells XiXiminus1 Among otheringredients its proof makes use of Theorem 57 consult [95 sect9] for details

Example 14 (Knorrer periodicity) Let q = fg+qprime where f g and qprime are forms ofdegrees a gt 0 b gt 0 and a+b in disjoint sets of variables (xi)i=1m (yj)j=1nand (zl)l=1p respectively Such a decomposition holds for example in the case ofisotropic quadratic forms q Let us write Q and Qprime for the projective hypersurfacesdefined by q and qprime respectively Assume that Q is smooth Under this assumptionwe have a Gm-action on Q given by λ middot (x y z) = (λbx λminusay z) with fixed point

locus Pmminus1qPnminus1qQprime this implies that Qprime is also smooth By combining Theorem13 and Example 12 with the fact that U(Pn) U(k)oplus(n+1) (see [81 sect242]) weobtain an induced isomorphism U(Q) U(k)oplus(m+n) oplus U(Qprime) Morally speakingthis shows that (modulo k) the additive invariants of Q and Qprime are the same

12 Orbifolds Let G be a finite group of order n (we assume that 1n isin k)ϕ the set of all cyclic subgroups of G ϕsim a set of representatives of the conju-gacy classes in ϕ X a smooth k-scheme equipped with a G-action and [XG]the associated orbifold As explained in [96 sect3] the assignment [V ] 7rarr V otimesk minuswhere V stands for a G-representation gives rise to an action of the representationring R(G) on U([XG]) Given σ isin ϕ let eσ be the unique idempotent of theZ[1n]-linearized representation ring R(σ)1n whose image under all the restric-tions R(σ)1n rarr R(σprime)1n with σprime ( σ is zero The normalizer N(σ) of σ actsnaturally on [Xσσ] and hence on U([Xσσ]) By functoriality this action restrictsto the direct summand eσU([Xσσ])1n

Theorem 15 ([96 Thm 11 and Cor 16]) The following computations hold(i) We have an induced isomorphism

(16) U([XG])1n oplusσisinϕsim

(eσU([Xσσ])1n)N(σ)

in the Z[1n]-linearized (and idempotent completed) category Hmo0(k)1n

(ii) If k contains the nth roots of unity then (16) reduces to an isomorphism


(U(Xσ)1n otimesZ[1n] eσR(σ)1n)N(σ)

where minus otimesZ[1n] minus stands for the canonical action of the category of finitelygenerated projective Z[1n]-modules on Hmo0(k)1n

(iii) If k contains the nth roots of unity and F is a field which contains the nth

roots of unity and 1n isin F then we have induced isomorphisms

(18) U([XG])F oplusgisinGsim

U(Xg)C(g)F (

oplusgisinG

U(Xg)F )G

in the category Hmo0(k)F where C(g) stands for the centralizer of gMoreover (17)-(18) are isomorphisms of (commutative) monoids

Roughly speaking Theorem 15 shows that the additive invariants of orbifoldscan be computed using solely ldquoordinaryrdquo schemes

4 GONCALO TABUADA

Example 19 (McKay correspondence) In many cases the dg category perfdg([XG])is known to be Morita equivalent to perfdg(Y ) for a crepant resolution Y of the(singular) geometric quotient XG see [8 15 42 44] This is generally referredto as the ldquoMcKay correspondencerdquo Whenever it holds we can replace [XG] byY in the formulas (16)-(18) Here is an illustrative example (with k algebraicallyclosed) the cyclic group G = C2 acts on any abelian surface S by the involutiona 7rarr minusa and the Kummer surface Km(S) is defined as the blow-up of SC2 at its16 singular points In this case the dg category perfdg([SC2]) is Morita equivalentto perfdg(Km(S)) Consequently Theorem 15(ii) leads to an isomorphism

(110) U(Km(S))12 U(S)C2

12 oplus U(k)oplus1612

Note that since the Kummer surface is Calabi-Yau the category perf(Km(S)) doesnot admit non-trivial semi-orthogonal decompositions This shows that the isomor-phism (110) is not induced from a semi-orthogonal decomposition

Corollary 111 (Algebraic K-theory) If k contains the nth roots of unity thenwe have the following isomorphism of Z-graded commutative Z[1n]-algebras

(112) Klowast([XG])1n oplusσisinϕsim

(Klowast(Xσ)1n otimesZ[1n] eσR(σ)1n)N(σ)

The formula (112) was originally established by Vistoli in [103 Thm 1] Amongother ingredients Vistolirsquos proof makes essential use of devissage The proof ofTheorem 15 and hence of (112) is not only different but moreover avoids the useof devissage consult [96 sect6] for details

Corollary 113 (Cyclic homology) If k contains the nth roots of unity then wehave the following isomorphisms of Z-graded commutative k-algebras

(114) HClowast([XG]) oplusgisinGsim

HClowast(Xg)C(g) (

oplusgisinG

HClowast(Xg))G

The formula (114) was originally established by Baranovsky in [4 Thm 11]Baranovskyrsquos proof is very specific to cyclic homology In constrast the proof ofTheorem 15 and hence of (114) avoids all the specificities of cyclic homology andis moreover quite conceptual consult [96 sect6] for details

Corollary 115 (Topological periodic cyclic homology) Let k be a perfect fieldof characteristic p gt 0 W (k) the associated ring of p-typical Witt vectors andK = W (k)[1p] the fraction field of W (k) If k contains the nth roots of unitythen we have the following isomorphisms of Z2-graded commutative K-algebras

(116) TPlowast([XG])1p oplusgisinGsim

TPlowast(Xg)C(g)1p (

oplusgisinG

TPlowast(Xg)1p)

G

To the best of the authorrsquos knowledge the formula (116) is new in the literatureconsult [96 sect1] for further corollaries of Theorem 15

121 Twisted analogues Given a sheaf of Azumaya algebras F over [XG] ie aG-equivariant sheaf of Azumaya algebras over X all the computations of Theorem15 admit F-twisted analogues consult [96 Thm 127 and Cor 129] for details


2 Noncommutative pure motives

In sect21 we recall the definition of the different categories of noncommutative puremotives Subsections sect22-24 are devoted to three structural properties of thesecategories (relation with the Brauer group semi-simplicity and rigidity) In 25 weprove to a far-reaching noncommutative generalization of the Weil conjectures seeTheorem 221 Finally in sect26 we describe some of the equivariant analogues of thetheory of noncommutative pure motives

21 Recollections Recall from [81 sect41] that the category of noncommutativeChow motives NChow(k) is defined as the idempotent completion of the full sub-category of Hmo0(k) consisting of the objects U(A) with A a smooth proper dgcategory By construction this category is additive rigid symmetric monoidal andcomes equipped with a symmetric monoidal functor U dgcatsp(k) rarr NChow(k)Moreover given smooth proper dg categories A and B we have isomorphisms

(21) HomNChow(k)(U(A) U(B)) K0(Dc(Aop otimes B)) = K0(Aop otimes B)

Given a rigid symmetric monoidal category (Cotimes1) consider the otimes-ideal

otimesnil(a b) = f isin HomC(a b) | fotimesn = 0 for some n 0

Recall from [81 sect44] that the category of noncommutative otimes-nilpotent motivesNVoev(k) is defined as the idempotent completion of the quotient NChow(k)otimesnil

As explained in [81 sect45] periodic cyclic homology gives rise to an additivesymmetric monoidal functor HPplusmn NChow(k) rarr VectZ2(k) with values in thecategory of finite-dimensional Z2-graded k-vector spaces Recall from loc citthat the category of noncommutative homological motives NHom(k) is defined asthe idempotent completion of the quotient NChow(k)Ker(HPplusmn)


N (a b) = f isin HomC(a b) | forallg isin HomC(b a) we have tr(g f) = 0

where tr(g f) stands for the categorical trace of the endomorphism g f Recallfrom [81 sect46] that the category of noncommutative numerical motives NNum(k)is defined as the idempotent completion of the quotient NChow(k)N

22 Relation with the Brauer group Let Br(k) be the Brauer group of thebase field k Given a central simple k-algebra A we write [A] for its Brauer class

Example 22 (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) Z2 Br(C) = 0 and Br(k) QZ in all the remainingcases Moreover every element of Br(k) can be represented by a cyclic k-algebra

Recall from [81 sect244] that we have the following equivalence

(23) [A] = [B]hArr U(A) U(B)

for any two central simple k-algebras A and B Intuitively speaking (23) showsthat the Brauer class [A] and the noncommutative Chow motive U(A) containexactly the same information Let K0(NChow(k)) be the Grothendieck ring of theadditive symmetric monoidal category NChow(k) Given a central simple k-algebraA we write [U(A)] for the Grothendieck class of U(A) The (proof of the) nextresult is contained in [80 Thm 612][88 Thm 13]

6 GONCALO TABUADA

Theorem 24 Given central simple k-algebras A and B we have the equivalence

(25) U(A) U(B)hArr [U(A)] [U(B)]

Roughly speaking Theorem 24 shows that the noncommutative Chow motivesof central simple k-algebras are insensitive to the Grothendieck group relations Bycombining the equivalences (23) and (25) we obtain the following result

Corollary 26 The following map is injective

Br(k) minusrarr K0(NChow(k)) [A] 7rarr [U(A)]

Consult sect511 for some applications of Corollary 26 to secondary K-theory

Remark 27 (Generalizations) Theorem 24 and Corollary 26 hold more generallywith k replaced by a base k-scheme X Furthermore instead of the Brauer groupBr(X) we can consider the second etale cohomology group1 H2

et(XGm) consult[80 88] for details In the case of an affine cone over a smooth irreducible planecomplex curve of degree ge 4 the latter etale cohomology group contains non-torsionclasses The same phenomenon occurs for example in the case of Mumfordrsquos(celebrated) singular surface [67 page 75] see [88 Example 132]

Remark 28 (Jacques Titsrsquo 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 ldquocut-and-pasterdquo relations Although very important the structureof this ring still remains poorly understood Among other ingredients Theorem24 was used in the construction of a new motivic measure microT entitled Tits motivicmeasure consult [87] for details This new motivic measure led to the proof ofseveral new structural properties of K0Var(k) For example making use of microT 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 it was proved in loc citthat two products of conics have the same Grothendieck class if and only if theyare isomorphic this refines a previous result of Kollar [49]

23 Semi-simplicity Let F be a field of characteristic zero The following resultis obtained by combining [81 Thm 427] with [93 Thm 11]

Theorem 29 The category NNum(k)F is abelian semi-simple

Assuming certain (polarization) conjectures Kontsevich conjectured in his sem-inal talk [53] that the category NNum(k)F was abelian semi-simple Theorem 29not only proves this conjecture but moreover shows that Kontsevichrsquos insight holdsunconditionally Let Num(k)F be the (classical) category of numerical motives see[1 sect4] The next result is obtained by combining [81 Rk 432] with [93 Cor 12]

Corollary 210 The category Num(k)F is abelian semi-simple

Assuming certain (standard) conjectures Grothendieck conjectured in the sixtiesthat the category Num(k)F was abelian semi-simple This conjecture was provedunconditionally by Jannsen [36] in the nineties using etale cohomology Corollary210 provides us with an alternative proof of Grothendieckrsquos conjecture

1As proved by Gabber [23] and de Jong [38] in the case where X admits an ample line bundle(egX affine) the Brauer group Br(X) may be identified with the torsion subgroup ofH2

et(XGm)


231 Numerical Grothendieck group The Grothendieck group K0(A) of a properdg category A comes equipped with the following Euler bilinear pairing

χ K0(A)timesK0(A) minusrarr Z ([M ] [N ]) 7rarrsumn

(minus1)ndimkHomDc(A)(MN [n])

This bilinear pairing is in general not symmetric neither skew-symmetric Never-theless when A is moreover smooth the associated left and right kernels of χ agreesee [81 Prop 424] Consequently under these assumptions on A we have a well-defined numerical Grothendieck group K0(A)simnum = K0(A)Ker(χ) Following[81 Thm 426] given smooth proper dg categories A and B we have isomorphisms

(211) HomNNum(k)(U(A) U(B)) K0(Aop otimes B)Ker(χ)

The next result whose proof makes use of Theorem 29 is obtained by combining[94 Thm 12] with [93 Thm 62]

Theorem 212 K0(A)simnum is a finitely generated free abelian group

Given a smooth proper k-scheme X let us write Zlowast(X)simnum for the (graded)group of algebraic cycles on X up to numerical equivalence By combining Theorem212 with the Hirzebruch-Riemann-Roch theorem we obtain the following result

Corollary 213 Zlowast(X)simnum is a finitely generated free abelian (graded) group

24 Rigidity Recall that a field extension lk is called primary if the algebraicclosure of k in l is purely inseparable over k When k is algebraically closed everyfield extension lk is primary

Theorem 214 ([90 Thm 21(i)]) Given a primary field extension lk and a fieldF of characteristic zero the base-change functor minusotimesk l NNum(k)F rarr NNum(l)Fis fully-faithful The same holds integrally when k is algebraically closed

Intuitively speaking Theorem 214 shows that the theory of noncommutativenumerical motives is ldquorigidrdquo under base-change along primary field extensionsAlternatively thanks to the isomorphisms (211) Theorem 214 shows that thenumerical Grothendieck group is ldquorigidrdquo under primary field extensions The com-mutative counterpart resp mixed analogue of Theorem 214 was established byKahn in [40 Prop 55] resp is provided by Theorem 617

Remark 215 (Extra functoriality) Let lk be a primary field extension As provedin [90 Thm 23] Theorems 29 and 214 imply that the base-change functor admitsa left=right adjoint Without the assumption that the field extension lk is primarysuch an adjoint functor does not exists in general consult [90 Rk 24] for details

25 Zeta functions of endomorphisms Let NM isin NChow(k)Q be a noncom-mutative Chow motive and f an endomorphism of NM Following Kahn [39Def 31] the zeta function of f is defined as the following formal power series

(216) Z(f t) = exp

sumnge1

tr(fn)tn

n

isin QJtK

where fn stands for the composition of f with itself n-times tr(fn) isin Q stands

for the categorical trace of fn and exp(t) =summge0

tm

m isin QJtK

8 GONCALO TABUADA

Remark 217 When NM = U(A)Q and f = [B]Q with B isin Dc(Aop otimes A) a dgA-A-bimodule (see sect21) we have the following computation

(218) tr(fn) = [HH(A BotimesLA middot middot middot otimesL

A B︸︷︷︸n-times

)] isin K0(k) Z

where HH(A B otimesLA middot middot middot otimesL

A B) stands for the Hochschild homology of A with co-efficients in B otimesL

A middot middot middot otimesLA B see [81 Prop 226] Intuitively speaking the integer

(218) is the ldquonumber of fixed pointsrdquo of the dg A-A-bimodule BotimesLA middot middot middot otimesL

A B

Example 219 (Zeta function) Let k = Fq be a finite field X a smooth properk-scheme and Fr the geometric Frobenius When A = perfdg(X) and B is thedg bimodule associated to the pull-back dg functor Frlowast perfdg(X) rarr perfdg(X)(218) reduces to [HH(X ΓFrn)] = 〈∆ middot ΓFrn〉 = |X(Fqn)| Consequently (216)

reduces to the (classical) zeta function ZX(t) = exp(sumnge1 |X(Fqn)| t

n

n ) of X

Remark 220 (Witt vectors) Recall from [32] the definition of the ring of (big)Witt vectors W(Q) = (1 + tQJtKtimes lowast) Since the leading term of (216) is equal to1 the zeta function Z(f t) of f belongs to W(Q) Moreover given endomorphismsf and f prime of noncommutative Chow motives NM and NM prime we have Z(f oplus f prime t) =Z(f t)times Z(f prime t) and Z(f otimes f prime t) = Z(f t) lowast Z(f prime t) in W(Q)

Let B =prodiBi be a finite-dimensional semi-simple Q-algebra Zi the center of

Bi δi for the degree [Zi Q] and di the index [Bi Zi]12 Given a unit b isin Btimes its

ith reduced norm Nrdi(b) isin Q is defined as the composition (NZiQ NrdBiZi)(bi)Let NM isin NChow(k)Q be a noncommutative Chow motive Thanks to Theorem

29 B = EndNNum(k)Q(NM) is a finite-dimensional semi-simple Q-algebra let uswrite ei isin B for the central idempotent corresponding to the summand Bi Givenan invertible endomorphism f of NM its determinant det(f) isin Q is defined as the

following productprodi Nrdi(f)microi where microi = tr(ei)

δidi

Theorem 221 ([93 Thm 58]) (i) The series Z(f t) isin QJtK is rational ie

Z(f t) = p(t)q(t) with p(t) q(t) isin Q[t] Moreover deg(q(t))minusdeg(p(t)) = tr(idNM )

(ii) When f is invertible we have the following functional equation

Z(fminus1 tminus1) = (minust)tr(idNM )det(f)Z(f t)

Corollary 222 (Weil conjectures) Let k = Fq be a finite field X a smooth properk-scheme X of dimension d and E = 〈∆ middot∆〉 isin Z the self-intersection number ofthe diagonal ∆ of X timesX(i) The zeta function ZX(t) of X is rational Moreover deg(q(t))minusdeg(p(t)) = E

(ii) We have the following functional equation ZX( 1qdt

) = plusmntEq d2 EZX(t)

Weil conjectured2 in [110] that the zeta function ZX(t) of X was rational andthat it satisfied a functional equation These conjectures were proved indepen-dently by Dwork [20] and Grothendieck [26] using p-adic analysis and etale coho-mology respectively Corollary 222 provides us with an alternative proof of theWeil conjectures see [93 Cor 512] Moreover Theorem 221 proves a far-reachingnoncommutative generalization of the Weil conjectures

2Weil conjectured also that the zeta function ZX(t) of X satisfied an analogue of the Riemannhypothesis This conjecture was proved by Deligne [18] using among other tools Lefschetz pencils


26 Equivariant noncommutative motives Let G be a finite group of order n(we assume that 1n isin k) Recall from [82 Def 41] the definition of a G-action ona dg category A Given a G-action G A we have an associated dg category AGof G-equivariant objects From a topological viewpoint AG may be understood asthe ldquohomotopy fixed pointsrdquo of the G-action on A Here are two examples

Example 223 (G-schemes) Given a G-scheme X the dg category perfdg(X) in-

herits a G-action In this case the dg category perfdg(X)G is Morita equivalent to

the dg category of G-equivariant perfect complexes perfGdg(X) = perfdg([XG])

Example 224 (Cohomology classes) Given a cohomology class [α] isin H2(G ktimes)the dg category k inherits a G-action G α k In this case the dg category ofG-equivariant objects is Morita 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 Asexplained in [82 sect5] the category NChow(k) admits a G-equivariant counter-

part NChowG(k) Recall from loc cit that the latter category is additiverigid symmetric monoidal and comes equipped with a symmetric monoidal functorUG dgcatGsp(k)rarr NChowG(k) Moreover we have isomorphisms

HomNChowG(k)(UG(G A) UG(G B)) KG

0 (Aop otimes B)

where the right-hand side stands for the G-equivariant Grothendieck group Inparticular the ring of endomorphisms of the otimes-unit UG(G 1 k) agrees with therepresentation ring3 R(G) Let us write I for the augmentation ideal associated tothe rank homomorphism R(G) Z

261 Relation with equivariant Chow motives Making use of Edidin-Grahamrsquoswork [21] on equivariant intersection theory Laterveer [59] and Iyer and Muller-Stach [35] 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)rarr ChowG(k)defined on smooth projective G-schemes

Theorem 225 ([82 Thm 84]) There exists a Q-linear fully-faithful symmetricmonoidal ΦGQ making the following diagram commute

SmProjG(k)X 7rarrGperfdg(X)

hG(minus)Q

dgcatGsp(k)

UG(minus)Q

ChowG(k)Q

NChowG(k)Q

(minus)IQ

ChowG(k)QminusotimesQ(1)ΦGQ

NChowG(k)QIQ

where ChowG(k)QminusotimesQ(1) stands for the orbit category of ChowG(k)Q with respectto the G-equivariant Tate motive Q(1) (see [81 sect42]) and (minus)IQ for the localizationfunctor associated to the augmentation ideal IQ

3Recall that when k = C and G is abelian we have an isomorphism R(G) Z[G]

10 GONCALO TABUADA

Roughly speaking Theorem 225 shows that in order to compare the equivariantcommutative world with the equivariant noncommutative world we need to ldquootimes-trivializerdquo the G-equivariant Tate motive Q(1) on one side and to localize at the aug-mentation ideal IQ on the other side Only after these two reductions the equivari-ant commutative world embeds fully-faithfully into the equivariant noncommuta-tive world As illustrated in sect262 this shows that the G-equivariant Chow motivehG(X)Q and the G-equivariant noncommutative Chow motive UG(G perfdg(X))contain (important) independent information about X

262 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 directionwe can associate to X the G-action G perfdg(X) The following result whoseproof makes use of Theorem 225 relates these two distinct directions of study

Theorem 226 ([82 Thm 12]) If the category perf(X) admits a full exceptionalcollection (E1 En) of G-invariant objects (6= G-equivariant objects) then thereexists a choice of integers r1 rn isin 0 dim(X) such that

(227) hG(X)Q Lotimesr1 oplus middot middot middot oplus Lotimesrn

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

Theorem 226 can be applied for example to any G-action on projective spacesquadrics Grassmannians etc consult [82 Examples 99-911] for details Morallyspeaking Theorem 226 shows that the existence of a full exceptional collection ofG-invariant objects completely determines the G-equivariant Chow motive hG(X)QIn particular hG(X)Q loses all the information about the G-action on X In con-trast as explained in [82 Rmk 94 and Prop 98] theG-invariant objects E1 Enyield (non-trivial) cohomology classes [α1] [αn] isin H2(G ktimes) such that

(228) UG(G perfdg(X)) UG(G α1k)oplus middot middot middot oplus UG(G αn k)

Taking into account (227)-(228) 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

3 Noncommutative (standard) conjectures

In sect31 we recall some important conjectures of Grothendieck Voevodsky andTate Subsection sect32 is devoted to their noncommutative counterparts As a firstapplication of the noncommutative viewpoint we prove that the original conjecturesof Grothendieck Voevodsky and Tate are invariant under homological projectiveduality This leads to a proof of these original conjectures in several new cases Asa second application we extend the original conjectures from schemes to algebraicstacks and prove them in the case of ldquolow-dimensionalrdquo orbifolds

31 Recollections Let k be a base field of characteristic zero Given a smoothproper k-scheme X and a Weil cohomology theory Hlowast let us write πnX for the nth

Kunneth projector of Hlowast(X) Zlowast(X)Q for the Q-vector space of algebraic cycles onX and Zlowast(X)Qsimnil Z

lowast(X)Qsimhom and Zlowast(X)Qsimnum for the quotient of Zlowast(X)Qwith respect to the smash-nilpotence homological and numerical equivalence rela-tion respectively Recall from [81 sect308-3011] that


(i) The Grothendieckrsquos standard conjecture4 of type C+ denoted by C+(X) as-serts that the even Kunneth projector π+

X =sumn π

2nX is algebraic

(ii) The Grothendieckrsquos standard conjecture of type D denoted by D(X) assertsthat Zlowast(X)Qsimhom = Zlowast(X)Qsimnum

(iii) The Voevodskyrsquos nilpotence conjecture V (X) (which implies Grothendieckrsquosconjecture D(X)) asserts that Zlowast(X)Qsimnil = Zlowast(X)Qsimnum

(iv) The Schur-finiteness conjecture5 denoted by S(X) asserts that the Chowmotive h(X)Q is Schur-finite in the sense of Deligne [17 sect1]

Remark 31 (Status) (i) Thanks to the work of Grothendieck and Kleiman (see[25 47 48]) the conjecture C+(X) holds when dim(X) le 2 and also forabelian varieties Moreover this conjecture is stable under products

(ii) Thanks to the work of Lieberman [62] the conjecture D(X) holds whendim(X) le 4 and also for abelian varieties

(iii) Thanks to the work Voevodsky [106] and Voisin [108] the conjecture V (X)holds when dim(X) le 2 Thanks to the work of Kahn-Sebastian [41] theconjecture V (X) holds moreover when X is an abelian 3-fold

(iv) Thanks to the work of Kimura [46] and Shermenev [74] the conjecture S(X)holds when dim(X) le 1 and also for abelian varieties

Let k = Fq be a finite base field of characteristic p gt 0 Given a smooth properk-scheme X and a prime number l 6= p recall from [97 98] that the Tate conjecturedenoted by T l(X) asserts that the cycle class map is surjective

Zlowast(X)Ql minusrarr H2lowastl-adic(XkQl(lowast))

Gal(kk)

Remark 32 (Status) Thanks to the work of Tate [98] the conjecture T l(X) holdswhen dim(X) le 1 and also for abelian varieties Thanks to the work of severalother people (consult Totarorsquos survey [102]) the conjecture T l(X) holds moreoverwhen X is a K3-surface (and p 6= 2)

32 Noncommutative counterparts Let k be a base field of characteristic zeroRecall from sect2 that periodic cyclic homology descends to the category of noncom-mutative Chow motives yielding a functor HPplusmn NChow(k)Q rarr VectZ2(k) Givena smooth proper dg category A consider the following Q-vector spaces

K0(A)Qsim = Hom(U(k)Q U(A)Q)

where belongs to nilhomnum and NVoev(k)QNHom(k)QNNum(k)Q re-spectively Under these notations the important conjectures in sect31 admit thefollowing noncommutative counterparts

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

A of HPplusmn(A) is algebraic

ie there exists an endomorphism π+A of U(A)Q such that HPplusmn(π+

A) = π+A

Conjecture Dnc(A) The equality K0(A)Qsimhom = K0(A)Qsimnum holds

Conjecture Vnc(A) The equality K0(A)Qsimnil = K0(A)Qsimnum holds

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

4The 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 πminus

X =sum

n π2n+1X is also algebraic

5Consult sect65 for the mixed analogue of the Schur-finiteness conjecture

12 GONCALO TABUADA

Let k = Fq be a finite base field of characteristic p gt 0 Given a smooth properdg category A and a prime number l 6= p consider the following abelian groups

Hom(Z(linfin) πminus1LKUK(AotimesFq Fqn)

)n ge 1 (33)

where Z(linfin) stands for the Prufer l-group and LKUK(A otimesk kn) for the Bousfieldlocalization of the algebraic K-theory spectrum K(AotimesFq Fqn) with respect to topo-logical complex K-theory KU Under these notations Tatersquos conjecture admits thefollowing noncommutative counterpart

Conjecture T lnc(A) The abelian groups (33) are zero

We now relate the conjectures in sect31 with their noncommutative counterparts

Theorem 34 Given a smooth proper k-scheme X we have the equivalences

C+(X) hArr C+nc(perfdg(X))(35)

D(X) hArr Dnc(perfdg(X))(36)

V (X) hArr Vnc(perfdg(X))(37)

S(X) hArr Snc(perfdg(X))(38)

T l(X) hArr T lnc(perfdg(X)) (39)

Morally speaking Theorem 34 shows that the important conjectures in sect31belong not only to the realm of algebraic geometry but also to the broad non-commutative setting of smooth proper dg categories Consult [81 sect5] and thereferences therein for the implications rArr in (35)-(36) and also for the equiva-lences (37)-(38) The converse implications lArr in (35)-(36) were established in[84 Thm 11] Finally the equivalence (39) was proved in [91 Thm 12]

321 Homological projective duality For a survey on homological projective du-ality (=HPD) we invite the reader to consult Kuznetsovrsquos ICM address [55] LetX be a smooth projective k-scheme equipped with a line bundle LX(1) we writeX rarr P(W ) for the associated morphism where W = H0(XLX(1))lowast Assume thatthe category perf(X) admits a Lefschetz decomposition 〈A0A1(1) Aiminus1(iminus1)〉with respect to LX(1) in the sense of [56 Def 41] Following [56 Def 61] let Ybe the HP-dual of X LY (1) the HP-dual line bundle and Y rarr P(W lowast) the mor-phism associated to LY (1) Given a linear subspace L sub W lowast consider the linearsections XL = X timesP(Wlowast) P(L) and YL = Y timesP(W ) P(Lperp) The next result whoseproof makes use of Theorem 34 is obtained by concatenating [81 sect53-54] with[84 Thm 14][85 Thm 11][91 Thm 13]

Theorem 310 (HPD-invariance6) Let X and Y be as above Assume that XL

and YL are smooth that dim(XL) = dim(X)minus dim(L) that dim(YL) = dim(Y )minusdim(Lperp) and that the following conjectures hold

(311) C+nc(A0dg) Dnc(A0dg) Vnc(A0dg) Snc(A0dg) T lnc(A0dg)

where A0dg stands for the dg enhancement of A0 induced by perfdg(X) Underthese assumptions we have the following equivalences of conjectures

(XL)hArr (YL) with isin C+ D V S T l

6Consult Theorem 78 for another HPD-invariance type result


Remark 312 The conjectures (311) hold for example whenever the triangulatedcategory A0 admits a full exceptional collection (this is the case in all the examplesin the literature) Furthermore Theorem 310 holds more generally when Y (or X)is singular In this case we need to replace Y by a noncommutative resolution ofsingularities in the sense of [55 sect24]

Theorem 310 shows that the conjectures in sect31 are invariant under homologicalprojective duality As a consequence we obtain the following practical result

Corollary 313 Let XL and YL be smooth linear sections as in Theorem 310(a) If dim(YL) le 2 then the conjectures C+(XL) and V (XL) hold(b) If dim(YL) le 4 then the conjecture D(XL) holds(c) If dim(YL) le 1 then the conjectures S(XL) and T l(XL) hold

By applying Corollary 313 to the Veronese-Clifford duality to the spinor dualityto the Grassmannian-Pfaffian duality to the determinantal duality and to other(incomplete) HP-dualities (see [55 sect4]) we obtain a proof of the conjectures insect31 in several new cases consult [7 84 85 91] for details In the particular caseof the Veronese-Clifford duality Corollary 313 leads furthermore to an alternativeproof of the Tate conjecture for smooth complete intersections of two quadrics (theoriginal (geometric) proof based on the notion of variety of maximal planes is dueto Reid [72]) consult [91 Thm 17] for details

322 Algebraic stacks Theorem 34 allows us to easily extend the important con-jectures in sect31 from smooth proper schemes to smooth proper algebraic stacks Xby setting (X ) =nc(perfdg(X )) where isin C+ D V S T l The next resultobtained by combining [96 Thm 92] with [91 Thm 19] proves these conjec-tures in the case of ldquolow-dimensionalrdquo orbifolds consult [84] for further examplesof algebraic stacks satisfying these conjectures

Theorem 314 Let G be a finite group X a smooth projective k-scheme equippedwith a G-action and X = [XG] the associated orbifold(a) The conjectures C+(X ) and V (X ) hold when dim(X) le 2 The conjecture

C+(X ) also holds when G acts by group homomorphisms on an abelian variety(b) The conjecture D(X ) holds when dim(X) le 4(c) The conjectures S(X ) and T l(X ) hold when dim(X) le 1

Roughly speaking Theorem 314 shows that the above conjectures are ldquoinsensi-tiverdquo to the G-action Among other ingredients its proof makes use of Theorem 15

Remark 315 (Generalizations) Theorem 314 holds more generally under the as-sumption that the conjectures in sect31 are satisfied by the fixed point locus Xσσwith σ isin ϕsim For example the conjecture T l(X ) also holds when X is an abeliansurface and the group G = C2 acts by the involution a 7rarr minusa

4 Noncommutative motivic Galois groups

Let F be a field of characteristic zero and NNum(k)F the abelian category ofnumerical motives The next result was proved in [81 Thm 64] and [93 Thm 71]

Theorem 41 The category NNum(k)F is super-Tannakian in the sense of Deligne[17] When F is algebraically closed NNum(k)F is neutral super-Tannakian

14 GONCALO TABUADA

By combining Theorem 41 with Delignersquos super-Tannakian formalism [17] weobtain an affine super-group F -scheme sGal(NNum(k)F ) called the noncommutativemotivic Galois super-group The following result relates this super-group with the(classical) motivic Galois super-group sGal(Num(k)F )

Theorem 42 ([93 Thm 74]) Assume that F is algebraically closed Then thereexists a faithfully flat morphism of affine super-group F -schemes

sGal(NNum(k)F ) Ker(sGal(Num(k)F )tlowast

Gm)

where Gm stands for the multiplicative (super-)group scheme and t for the inclusionof the category of Tate motives into numerical motives

Theorem 42 was envisioned by Kontsevich see his seminal talk [53] Intuitivelyspeaking it shows that the ldquootimes-symmetriesrdquo of the commutative world which can belifted to the noncommutative world are precisely those which become trivial whenrestricted to Tate motives Theorem 42 also holds when F is not algebraicallyclosed However in this case the super-group schemes are only defined over a (verybig) commutative F -algebra

Remark 43 (Simplification) The analogue of Theorem 42 with k of characteristiczero was proved in [81 Thm 67(ii)] However therein we assumed the noncom-mutative counterparts of the standard conjectures of type C+ and D and moreoverused Deligne-Milnersquos theory of Tate-triples In contrast Theorem 42 is uncondi-tional and its proof avoids the use of Tate-triples consult [93 sect7] for details

Base-change Recall from [81 sect6] the definition of the (conditional7) noncommu-

tative motivic Galois group Gal(NNumdagger(k)F )

Theorem 44 ([90 Thm 22]) Given a primary field extension lk the induced

base-change functor minus otimesk l NNumdagger(k)F rarr NNumdagger(l)F gives rise to a faithfully

flat morphism of affine group F -schemes Gal(NNumdagger(l)F )rarr Gal(NNumdagger(k)F )

Roughly speaking Theorem 42 shows that every ldquootimes-symmetryrdquo of the categoryof noncommutative numerical k-linear motives can be extended to a ldquootimes-symmetryrdquoof the category of noncommutative l-linear motives Among other ingredientsits proof makes use of Theorem 214 In the particular case of an extension ofalgebraically closed fields lk the commutative counterpart of Theorem 44 wasestablished by Deligne-Milne in [19 Prop 622(b)]

5 Localizing invariants

Recall from [81 sect81] the notion of a short exact sequence of dg categories inthe sense of DrinfeldKeller In sect51 we describe a key structural property of theseshort exact sequences and explain its implications to secondary K-theory

Recall from [81 sect851] the construction of the universal localizing A1-homotopyinvariant of dg categories U dgcat(k)rarr NMot(k) in loc cit we used the explicit

notation UA1

loc dgcat(k) rarr NMotA1

loc(k) In [81 sect853] we described the behaviorof U with respect to dg orbit categories and dg cluster categories In sect52-54 wedescribe the behavior of U with respect to openclosed scheme decompositions cor-ner skew Laurent polynomial algebras and noncommutative projective schemes Asexplained in [81 Thm 825] all the results in sect52-54 are motivic in the sense that

7We assume the noncommutative counterparts of the standard conjectures of type C+ and D


they hold similarly for every localizing A1-homotopy invariant such as mod-n alge-braic K-theory (when 1n isin k) homotopy K-theory etale K-theory periodic cyclichomology8 (when char(k) = 0) etc The results of sect54 do not require A1-homotopyinvariance and so they hold for every localizing invariant see Remark 526


51 Short exact sequences Recall from [81 sect84] the notion of a split short exactsequence of dg categories 0 rarr A rarr B rarr C rarr 0 Up to Morita equivalence thisdata is equivalent to inclusions of dg categories A C sube B yielding a semi-orthogonaldecomposition of triangulated categories H0(B) = 〈H0(A)H0(C)〉 in the sense ofBondal-Orlov [13] by definition the category H0(A) has the same objects as A andmorphisms H0(A)(x y) = H0(A(x y))

Theorem 52 ([80 Thm 44]) Let 0rarr Ararr B rarr C rarr 0 be a short exact sequenceof dg categories in the sense of DrinfeldKeller If A is smooth and proper and Bis proper then the short exact sequence is split

Morally speaking Theorem 52 shows that the smooth proper dg categories be-have as ldquoinjectiverdquo objects In the setting of triangulated categories this conceptualidea goes back to the pioneering work of Bondal-Kapranov [11]

511 Secondary K-theory Two decades ago Bondal-Larsen-Lunts introduced in[12] the Grothendieck ring of smooth proper dg categories PT (k) This ring isdefined by generators and relations The generators are the Morita equivalenceclasses of smooth proper dg categories9 and the relations [B] = [A] + [C] arise fromsemi-orthogonal decompositions H0(B) = 〈H0(A)H0(C)〉 The multiplication lawis induced by the tensor product of dg categories One decade ago Toen intro-duced in [101] a ldquocategorifiedrdquo 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 rarr A rarr B rarr C rarr 0The multiplication law is also induced by the tensor product of dg categories

Theorem 52 directly leads to the following result

Corollary 53 The rings PT (k) and K(2)0 (k) are isomorphic

Morally speaking Corollary 53 shows that the secondary Grothendieck ring isnot a new mathematical notion

By construction the universal additive invariant U (see sect1) sends semi-orthogonaldecompositions to direct sums Therefore it gives rise to a ring homomorphismPT (k)rarr K0(NChow(k)) Making use of Corollary 26 we then obtain the result


Br(k) minusrarr PT (k) K(2)0 (k) [A] 7rarr [A] (55)

8Periodic cyclic homology is not a localizing A1-homotopy invariant in the sense of [81 sect85]because it does not preserves filtered (homotopy) colimits Nevertheless all the results of sect52-54hold similarly for periodic cyclic homology

9Bondal-Larsen-Lunts worked originally with (pretriangulated) dg categories In this general-ity the classical Eilenbergrsquos swindle argument implies that the Grothendieck ring is trivial

16 GONCALO TABUADA

The map (55) may be understood as the ldquocategorificationrdquo of the canonical mapfrom the Picard group Pic(k) to the Grothendieck ring K0(k) In contrast withPic(k) rarr K0(k) the map (55) does not seems to admit a ldquodeterminantrdquo map inthe converse direction Nevertheless Corollary 54 shows that this map is injective

Remark 56 (Generalizations) Similarly to Remark 27 Corollaries 53-54 holdmore generally with k replaced by a base k-scheme X

52 Gysin triangle Let X be a smooth k-scheme i Z rarr X a smooth closedsubscheme and j V rarr X the open complement of Z

Theorem 57 ([95 Thm 19]) We have an induced distinguished ldquoGysinrdquo triangle

(58) U(Z)U(ilowast)minusrarr U(X)

U(jlowast)minusrarr U(V )partminusrarr U(Z)[1]

where ilowast resp jlowast stands for the push-forward resp pull-back dg functor

Remark 59 (Generalizations) As explained in [95 sect7] Theorem 57 holds not onlyfor smooth schemes but also for smooth algebraic spaces in the sense of Artin

Roughly speaking Theorem 57 shows that the difference between the localizingA1-homotopy invariants of X and of V is completely determined by the closedsubscheme Z Consult Remark 65 resp 69 for the relation between (58) andthe motivic Gysin triangles constructed by Morel-Voevodsky resp Voevodsky

521 Quillenrsquos localization theorem HomotopyK-theory is a localizing A1-homotopyinvariant which agrees with Quillenrsquos algebraic K-theory when restricted to smoothk-schemes Therefore Theorem 57 leads to the K-theoretical localization theorem

(510) K(Z)K(ilowast)minusrarr K(X)

K(jlowast)minusrarr K(V )partminusrarr K(Z)[1]

originally established by Quillen in [70 Chapter 7 sect3] Among other ingredientsQuillenrsquos proof makes essential use of devissage The proof of Theorem 57 andhence of (510) is quite different and avoids the use of devissage

522 Six-term exact sequence in de Rham cohomology Periodic cyclic homologyis a localizing A1-homotopy invariant (when char(k) = 0) Moreover thanks to theHochschild-Kostant-Rosenberg theorem we have an isomorphism of Z2-graded k-vector spaces HPplusmn(X) (

oplusn evenH

ndR(X)

oplusn oddH

ndR(X)) where HlowastdR stands

for de Rham cohomology Furthermore the maps i and j give rise to homomor-phisms Hn

dR(ilowast) HndR(Z) rarr Hn+2c

dR (X) and HndR(jlowast) Hn

dR(X) rarr HndR(V ) where

c stands for the codimension of i Therefore Theorem 57 leads to the followingsix-term exact sequence in de Rham cohomologyoplus

n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo

This exact sequence is the ldquo2-periodizationrdquo of the Gysin long exact sequence onde Rham cohomology originally constructed by Hartshorne in [31 Chapter II sect3]


523 Reduction to projective schemes As a byproduct of Theorem 57 the studyof the localizing A1-homotopy invariants of smooth k-schemes can be reduced tothe study of the localizing A1-homotopy invariants of smooth projective k-schemes

Theorem 511 ([95 Thm 21]) Let X a smooth k-scheme(i) If char(k) = 0 then U(X) belongs to the smallest triangulated subcategory of

NMot(k) containing the objects U(Y ) with Y a smooth projective k-scheme(ii) If k is a perfect field of characteristic p gt 0 then U(X)1p belongs to the

smallest thick triangulated subcategory of NMot(k)1p containing the objectsU(Y )1p with Y a smooth projective k-scheme

Among other ingredients the proof of item (i) resp item (ii) of Theorem 511makes use of resolution of singularities resp of Gabberrsquos refined version of deJongrsquos theory of alterations consult [95 sect8] for details

Remark 512 (Dualizable objects) Given a smooth projective k-scheme Y theassociated dg category perfdg(Y ) is smooth and proper see [81 Example 142]

Therefore since the universal localizing A1-homotopy invariant U is symmetricmonoidal it follows from [81 Thm 143] that U(Y ) is a dualizable object of thesymmetric monoidal category NMot(k) Given a smooth k-scheme X the asso-ciated dg category perfdg(X) is smooth but not necessarily proper NeverthelessTheorem 511 implies that U(X) resp U(X)1n is still a dualizable object of thesymmetric monoidal category NMot(k) resp NMot(k)1p

53 Corner skew Laurent polynomial algebras Let A be a unital k-algebrae an idempotent of A and φ A

simrarr eAe a ldquocornerrdquo isomorphism The associatedcorner skew Laurent polynomial algebra A[t+ tminusφ] is defined as follows the el-ements are formal expressions tmminusaminusm + middot middot middot + tminusaminus1 + a0 + a1t+ middot middot middot + ant

n+ with

aminusi isin φi(1)A and ai isin Aφi(1) for every i ge 0 the addition is defined component-wise the multiplication is determined by the distributive law and by the relationstminust+ = 1 t+tminus = e atminus = tminusφ(a) for every a isin A and t+a = φ(a)t+ for everya isin A Note that A[t+ tminusφ] admits a canonical Z-grading with deg(tplusmn) = plusmn1 Asproved in [2 Lem 24] the corner skew Laurent polynomial algebras can be charac-terized as those Z-graded algebras C =

oplusnisinZ Cn containing elements t+ isin C1 and

tminus isin Cminus1 such that tminust+ = 1 Concretely we have C = A[t+ tminusφ] with A = C0e = t+tminus and φ C0 rarr t+tminusC0t+tminus given by c0 7rarr t+c0tminus

Example 513 (Skew Laurent polynomial algebras) When e = 1 A[t+ tminusφ] 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 tminus1]

Example 514 (Leavitt algebras) Following [60] the Leavitt algebra Ln n ge 0 isthe k-algebra generated by elements x0 xn y0 yn subject to the relationsyixj = δij and

sumni=0 xiyi = 1 Note that the canonical Z-grading with deg(xi) = 1

and deg(yi) = minus1 makes Ln into a corner skew Laurent polynomial algebra Notealso that L0 k[t tminus1] In the remaining cases n ge 1 Ln is the universal example

of a k-algebra of module type (1 n+ 1) ie Ln Loplus(n+1)n as right Ln-modules

Example 515 (Leavitt path algebras) Let Q = (Q0 Q1 s r) be a finite quiverwith no sources Q0 and Q1 stand for the sets of vertices and arrows respectivelyand s and r for the source and target maps respectively Consider the doublequiver Q = (Q0 Q1 cup Qlowast1 s r) obtained from Q by adding an arrow αlowast in the

18 GONCALO TABUADA

converse direction for each arrow α isin Q1 The Leavitt path algebra LQ of Q is the

quotient of the quiver algebra kQ (which is generated by elements α isin Q1 cup Qlowast1and ei with i isin Q0) by the Cuntz-Kriegerrsquos relations αlowastβ = δαβer(α) for everyα β isin Q1 and

sumαisinQ1|s(α)=i αα

lowast = ei for every non-sink i isin Q0 Note that LQadmits a canonical Z-grading with deg(α) = 1 and deg(αlowast) = minus1 For every vertexi isin Q0 choose an arrow αi such that r(αi) = i and consider the associated elementst+ =

sumiisinQ0

αi and tminus = tlowast+ Since deg(tplusmn) = plusmn1 and tminust+ = 1 LQ is an exampleof a corner skew Laurent polynomial algebra In the particular case where Q is thequiver with one vertex and n+ 1 arrows LQ reduces to the Leavitt algebra Ln

Theorem 516 ([86 Thm 31]) We have an induced distinguished triangle

U(A)idminusU(φA)minusrarr U(A) minusrarr U(A[t+ tminusφ])

partminusrarr U(A)[1]

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

Roughly speaking Theorem 516 shows that A[t+ tminusφ] may be considered as amodel for the orbits of the N-action on U(A) induced by the endomorphism U(φA)

531 Leavitt path algebras Let Q = (Q0 Q1 s r) be a quiver as in Example 515with v vertices and vprime sinks Assume that the set Q0 is ordered with the first vprime

elements corresponding to the sinks Let I primeQ be the incidence matrix of Q IQ the

matrix obtained from I primeQ by removing the first vprime rows (which are zero) and ItQthe transpose of IQ Under these notations Theorem 516 (concerning the Leavittpath algebra LQ) admits the following refinement


(518) U(k)(vminusvprime)oplus ( 0id)minusI

tQminusrarr U(k)voplus minusrarr U(LQ)

partminusrarr U(k)[1](vminusvprime)oplus

Roughly speaking Theorem 517 shows that all the information about the lo-calizing A1-homotopy invariants of Leavitt path algebras LQ is encoded in theincidence matrix of the quiver Q As an application Theorem 517 directly leadsto the following explicit model for the mod-n Moore construction10

Example 519 (mod-n Moore construction) Let Q be the quiver with one vertexand n+1 arrows In this particular case (518) reduces to the distinguished triangle

U(k)nmiddotidminusrarr U(k) minusrarr U(Ln)

partminusrarr U(k)[1]

This shows that the Leavitt algebra Ln n ge 2 is a model for the mod-n Mooreobject of U(k) Therefore since the universal localizing A1-homotopy invariant Uis symmetric monoidal given a small dg category A we conclude that the tensorproduct Aotimes Ln is a model for the mod-n Moore object of U(A)

54 Noncommutative projective schemes Let A =oplus

nge0An be a N-gradedNoetherian k-algebra In what follows we assume that A is connected ie A0 = kand locally finite-dimensional ie dimk(An) ltinfin for every n Following Manin [63]Gabriel [24] Artin-Zhang [3] and others the noncommutative projective schemeqgr(A) associated to A is defined as the quotient abelian category gr(A)tors(A)where gr(A) stands for the abelian category of finitely generated Z-graded (right)

10Explicit models for the suspension construction namely the Waldhausenrsquos Sbull-constructionand the Calkin algebra are described in [81 sect832 and sect844]


A-modules and tors(A) for the Serre subcategory of torsion A-modules This def-inition was motivated by Serrersquos celebrated result [73 Prop 78] which assertsthat in the particular case where A is commutative and generated by elementsof degree 1 the quotient category qgr(A) is equivalent to the abelian category ofcoherent OProj(A)-modules coh(Proj(A)) For example when A is the polynomialk-algebra k[x1 xd] with deg(xi) = 1 we have the equivalence of categoriesqgr(k[x1 xd]) coh(Pdminus1) For a survey on noncommutative projective geom-etry we invite the reader to consult Staffordrsquos ICM address [76]

Assume that A is Koszul and has finite global dimension d Under these as-sumptions the Hilbert series hA(t) =

sumnge0 dimk(An)tn isin Z[[t]] is invertible and

its inverse hA(t)minus1 is a polynomial 1 minus β1t + β2t2 minus middot middot middot + (minus1)dβdt

d of degree d

with βi the dimension of the k-vector space TorAi (k k) (or ExtiA(k k))

Example 520 (Quantum polynomial algebras) Choose constant elements qij isin ktimeswith 1 le i lt j le d Following Manin [64 sect1] the N-graded Noetherian k-algebra

A = k〈x1 xd〉〈xjxi minus qijxixj | 1 le i lt j le d〉

with deg(xi) = 1 is called the quantum polynomial algebra associated to qij Thisalgebra is Koszul has global dimension d and hA(t)minus1 = (1minus t)d

Example 521 (Quantum matrix algebras) Choose a constant element q isin ktimesFollowing Manin [64 sect1] the N-graded Noetherian k-algebra A defined as thequotient of k〈x1 x2 x3 x4〉 by the following relations

x1x2 = qx2x1 x1x3 = qx3x1 x1x4 minus x4x1 = (q minus qminus1)x2x3

x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3

with deg(xi) = 1 is called the quantum matrix algebra associated to q This algebrais Koszul has global dimension 4 and hA(t)minus1 = (1minus t)4

Example 522 (Sklyanin algebras) Let C be a smooth elliptic k-curve σ an auto-morphism of C given by translation under the group law and L a line bundle onC of degree d ge 3 We write Γσ sub C times C for the graph of σ and W for the d-dimensional k-vector space H0(CL) Following Feigin-Odesskii [22] and Tate-Vanden Bergh [99 sect1] the N-graded Noetherian k-algebra A = T (W )R where

R = H0(C times C (L L)(minusΓσ)) sub H0(C times CL L) = W otimesW

is called the Sklyanin algebra associated to the triple (C σL) This algebra isKoszul has global dimension d and hA(t)minus1 = (1minus t)d

Example 523 (Homogenized enveloping algebras) Let g be a finite-dimensional Liealgebra Following Smith [75 sect12] the N-graded Noetherian k-algebra

A = T (goplus kz)〈z otimes xminus xotimes z |x isin g cup xotimes y minus y otimes xminus [x y]otimes z |x y isin g〉

is called the homogenized enveloping algebra of g This algebra is Koszul has globaldimension d = dim(g) + 1 and hA(t)minus1 = (1minus t)d

Given a N-graded k-algebraA as above let us writeDbdg(qgr(A)) for the canonical

dg enhancement of the bounded derived category of qgr(A) This dg category is ingeneral not proper see [89 sect1] The following result is contained [89 Thm 12]

20 GONCALO TABUADA

Theorem 524 We have an induced distinguished triangle

(525)

+infinoplusminusinfin

U(k)Mminusrarr


U(k) minusrarr U(Dbdg(qgr(A)))partminusrarr


U(k)[1]

where M stands for the (infinite) matrix Mij = (minus1)j(minus1)(iminusj)βiminusj Moreoverwhen βd = 1 the triangle (525) induces an isomorphism U(Dbdg(qgr(A))) U(k)oplusd

As proved in [78 Cor 02] we have hA(t)minus1 = (1minus t)3 whenever d = 3

Remark 526 (Localizing invariants) The proof of Theorem 524 does not makes useof A1-homotopy invariance Consequently as explained in [81 Thm 85] Theorem524 holds similarly for every localizing invariant in the sense of [81 Def 83] Ex-amples of localizing invariants which are not A1-homotopy invariant include noncon-nective algebraic K-theory Hochschild homology cyclic homology negative cyclichomology periodic cyclic homology (when char(k) = p gt 0) topological Hochschildhomology topological cyclic homology topological periodic cyclic homology etc

Roughly speaking Theorem 524 (and Remark 526) shows that the localizing in-variants of a noncommutative projective scheme qgr(A) are completely determinedby the Hilbert series hA(t)

6 Noncommutative mixed motives

In this section we assume that the base field k is perfect Kontsevich introducedin [50 51 53] a certain rigid symmetric monoidal triangulated category of non-commutative mixed motives NMix(k) As explained in [81 sect911] this categorycan be (conceptually) described as the smallest thick triangulated subcategory ofNMot(k) (see sect5) containing the objects U(A) with A smooth and proper

In sect61 we compute the Picard group of the thick triangulated subcategory ofNMix(k) generated by the noncommutative mixed motives of central simple k-algebras Subsections sect62-64 are devoted to the precise relation between the cat-egory NMix(k) and Morel-Voevodskyrsquos stable A1-homotopy category Voevodskyrsquostriangulated category of geometric mixed motives and Levinersquos triangulated cat-egory of mixed motives respectively In sect65 we address the Schur-finiteness con-jecture in the case of quadric fibrations Finally subsection sect66 is devoted to therigidity property of the category of mod-n noncommutative mixed motives

61 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 this major challenge can be met if we re-strict ourselves to central simple k-algebras Let NMixcsa(k) be the thick trian-gulated subcategory of NMix(k) generated by the noncommutative mixed mo-tives U(A) of central simple k-algebras A Similarly to sect22 the equivalence[A] = [B] hArr U(A) U(B) holds for any two central simple k-algebras A andB Moreover following [81 Thm 828] we have non-trivial Ext-groups

(61) HomNMix(k)(U(A)U(B)[minusn]) Kn(Aop otimesB) n isin 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 62 ([14 Thm 222]) We have the following isomorphism

Br(k)times Z simminusrarr Pic(NMixcsa(k)) ([A] n) 7rarr U(A)[n]


Theorem 62 shows that although NMixcsa(k) contains information about all thehigher algebraic K-theory of central simple k-algebras none of the noncommutativemixed motives which are built using the non-trivial Ext-groups (61) is otimes-invertible

62 Morel-Voevodskyrsquos motivic category Morel-Voevodsky introduced in [66105] the stable A1-homotopy category of (P1infin)-spectra SH(k) By construc-tion we have a symmetric monoidal functor Σinfin(minus+) Sm(k)rarr SH(k) defined onsmooth k-schemes Let KGL isin SH(k) be the ring (P1infin)-spectrum representinghomotopy K-theory and Mod(KGL) the homotopy category of KGL-modules

Theorem 63 (i) If char(k) = 0 then there exists a fully-faithful symmetricmonoidal triangulated functor Ψ making the following diagram commute

(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)

Ψ NMix(k)oplus NMot(k)

where Hom(minusminus) stands for the internal-Hom of the closed symmetric monoidalcategory NMot(k) (minus)or for the (contravariant) duality functor and NMix(k)oplus

for the smallest triangulated subcategory of NMot(k) which contains NMix(k)and is stable under arbitrary direct sums

(ii) If char(k) = p gt 0 then there exists a Z[1p]-linear fully-faithful symmetricmonoidal triangulated functor Ψ1p making the following diagram commute

Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p

NMix(k)oplus1p NMot(k)1p

Intuitively speaking Theorem 63 shows that as soon as we pass to KGL-modules the commutative world embeds fully-faithfully into the noncommutativeworld Consult [81 sect94] and the references therein for the construction of thetwo outer commutative diagrams The inner commutative squares follow from thecombination of Theorem 511 with Remark 512 consult [95 Thm 31] for details

Remark 65 (Morel-Voevodskyrsquos motivic Gysin triangle) Let X be a smooth k-scheme Z rarr X a smooth closed subscheme with normal vector bundle N andj V rarr X the open complement of Z Making use of homotopy purity Morel-Voevodsky constructed in [66 sect32][105 sect4] a motivic Gysin triangle

(66) Σinfin(V+)Σinfin(j+)minusrarr Σinfin(X+) minusrarr Σinfin(Th(N))

partminusrarr Σinfin(V+)[1]

22 GONCALO TABUADA

in SH(k) where Th(N) stands for the Thom space of N Since homotopy K-theoryis an orientable and periodic cohomology theory Σinfin(Th(N))andKGL is isomorphicto Σinfin(Z+) and KGL Using the commutative diagram (64) we hence observe thatthe image of (66) under the composed functor Ψ (minusandKGL) SH(k)rarr NMix(k)oplus

agrees with the dual of the Gysin triangle (58) In other words the Gysin triangle(58) is the dual of the ldquoKGL-linearizationrdquo of (66)

63 Voevodskyrsquos motivic category Voevodsky introduced in [104 sect2] the tri-angulated category of geometric mixed motives DMgm(k) By construction thiscategory comes equipped with a symmetric monoidal functorM Sm(k)rarr DMgm(k)and is the natural setting for the study of algebraic cycle (co)homology theories suchas higher Chow groups Suslin homology motivic cohomology etc

Theorem 67 There exists a Q-linear fully-faithful symmetric monoidal functorΦQ making the following diagram commute

(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)

DMgm(k)QminusotimesQ(1)[2] ΦQ

NMix(k)Q NMot(k)Q

Intuitively speaking Theorem 67 shows that as soon as we ldquootimes-trivializerdquo theTate motive Q(1)[2] the commutative world embedds fully-faithfully into the non-commutative world Consult [81 sect95] and the references therein for the construc-tion of the outer commutative diagram The inner commutative square follows fromthe combination of Theorem 511 with Remark 512 consult [95 Thm 37]

Remark 69 (Voevodskyrsquos motivic Gysin triangle) Let X be a smooth k-schemeZ rarr X a smooth closed subscheme of codimension c and j V rarr X the opencomplement of Z Making use of deformation to the normal cone Voevodskyconstructed in [104 sect2] a motivic Gysin triangle

(610) M(V )QM(j)Qminusrarr M(X)Q minusrarrM(Z)Q(c)[2c]

partminusrarrM(V )Q[1]

in DMgm(k)Q Using the commutative diagram (68) we observe that the imageof (610) under the (composed) functor ΦQ DMgm(k)Q rarr NMix(k)Q agrees withthe dual of the rationalized Gysin triangle (58) In other words the rationalizedGysin triangle (58) is the dual of the ldquoTate otimes-trivializationrdquo of (610)

Let DMetgm(k) be the etale variant of DMgm(k) introduced by Voevodsky in [104

sect33] As proved in loc cit DMgm(k)Q is equivalent to DMetgm(k)Q Consequently

Theorem 67 leads to the following result (see [95 Thm 313])

Corollary 611 (Etale descent) The presheaf of noncommutative mixed motivesSm(k)op rarr NMot(k)Q X 7rarr U(X)Q satisfies etale descent ie for every etale coverV = Vi rarr XiisinI of X we have an isomorphism U(X)Q holimnge0U(CnV)Q

where CbullV stands for the Cech simplicial k-scheme associated to the cover V


64 Levinersquos motivic category Levine introduced in [61 Part I] a triangu-lated category of mixed motives DM(k) and a (contravariant) symmetric monoidalfunctor h Sm(k) rarr DM(k) As proved in [34 Thm 42] the following assign-ment h(X)Q(n) 7rarr Hom(M(X)Q(n)) gives rise to an equivalence of categoriesDM(k)Q rarr DMgm(k)Q whose precomposition with h(minus)Q is X 7rarr M(X)orQ Con-sequently thanks to Theorem 67 there exists a Q-linear fully-faithful symmetricmonoidal functor ΦQ making the following diagram commute

(612) Sm(k)

h(minus)Q

X 7rarrperfdg(X) dgcat(k)

U(minus)Q

DM(k)Q

DM(k)QminusotimesQ(1)[2] ΦQ

NMix(k)Q NMot(k)Q

Note that in contrast with the diagrams of Theorems 63 and 67 the commutativediagram (612) does not uses any kind of duality functor

65 Schur-finiteness conjecture Given a smooth k-schemeX the Schur-finitenessconjecture denoted by S(X) asserts that the mixed motive M(X)Q is Schur-finitein the sense of Deligne [17 sect1] Thanks to the (independent) work of Guletskii [27]and Mazza [65] the conjecture S(X) holds when dim(X) le 1 and also for abelianvarieties In addition to these cases it remains wide open

Theorem 613 ([85 Thm 11]) Let q Qrarr B a flat quadric fibration of relativedimension d minus 2 Assume that B and Q are k-smooth and that q has only simpledegenerations ie that all the fibers of q have corank le 1 and that the locus D sub Bof the critical values of q is k-smooth Under these assumptions the following holds

(i) If d is even then we have S(B) + S(B) hArr S(Q) where B stands for thediscriminant 2-fold cover of B (ramified over D)

(ii) If d is odd and char(k) 6= 2 then we have S(Vi)+ S(Di) rArr S(Q) where

Vi is any affine open of B and Di is any Galois 2-fold cover of Di = D cap Vi

Roughly speaking Theorem 613 relates the Schur-finiteness conjecture for thetotal space Q with the Schur-finiteness conjecture for certain coveringssubschemesof the base B Among other ingredients its proof makes use of Theorem 67 and ofthe twisted analogue of Theorem 15 (see sect121) Theorem 613 enables the proofof the Schur-finiteness conjecture in the following new cases

Corollary 614 ([92 Cor 13 and 15]) Let q Qrarr B be as in Theorem 613(i) Assume that B is a curve and that char(k) 6= 2 when d is odd Under these

assumptions the conjecture S(Q) holds(ii) Assume that B is a surface that d is odd that char(k) 6= 2 and that the con-

jecture S(B) holds Under these assumptions the conjecture S(Q) also holds

Corollary 614(ii) can be applied for example to the case where B is an opensubscheme of an abelian surface or smooth projective surface with pg = 0 satisfyingBlochrsquos conjecture (see Guletskii-Pedrini [28 sect4 Thm 7]) Recall that Blochrsquosconjecture holds for surfaces not of general type (see Bloch-Kas-Leiberman [10])

24 GONCALO TABUADA

for surfaces which are rationally dominated by a product of curves (see Kimura[46]) for Godeaux Catanese and Barlow surfaces (see Voisin [107 109]) etc

Remark 615 (Bass-finiteness conjecture) Let k = Fq be a finite field and X asmooth k-scheme The Bass-finiteness conjecture (see [5 sect9]) asserts that the alge-braic K-theory groups Kn(X) n ge 0 are finitely generated Thanks to the workof Quillen [29 69 71] the Bass-finiteness conjecture holds when dim(X) le 1

In the same vein we can consider the mod 2-torsion Bass-finiteness conjecturewhere Kn(X) is replaced by Kn(X)12 As proved in [92] Theorem 613 andCorollary 614 hold similarly with the Schur-finiteness conjecture replaced by themod 2-torsion Bass-finiteness conjecture As a consequence we obtain a proof ofthe (mod 2-torsion) Bass-finiteness conjecture in new cases

66 Rigidity Given an integer n ge 2 recall from [81 sect99] the definition of thecategory of mod-n noncommutative mixed motives NMix(kZn) By constructiongiven smooth proper dg categories A and B we have isomorphisms

HomNMix(kZn)(U(A)U(B)[minusn]) Kn(Aop otimes BZn) n isin Z (616)

where the right-hand side stands for mod-n algebraic K-theory

Theorem 617 ([90 Thm 21(ii)]) Given an extension of separably closed fieldslk the base-change functor minusotimesk l NMix(kZn)rarr NMix(lZn) is fully-faithfulwhenever n is coprime to the characteristic of k

Theorem 617 is the mixed analogue of Theorem 214 Intuitively speaking itshows that the theory of mod-n noncommutative mixed motives is ldquorigidrdquo under ex-tensions of separably closed fields Alternatively thanks to the isomorphisms (616)Theorem 617 shows that mod-n algebraic K-theory is ldquorigidrdquo under extensions ofseparably closed fields This is a far-reaching noncommutative generalization ofSuslinrsquos celebrated rigidity theorem [79] consult [90 sect2] for details and also for ap-plications to equivariant and twisted algebraic K-theory In the particular case ofan extension of algebraically closed fields the commutative counterpart of Theorem617 was established by Haesemeyer-Hornbostel in [30 Thm 30]

7 Noncommutative realizations and periods

In this section we assume that the base field k is perfect Subsection sect71 is de-voted to the noncommutative realizations associated to the (classical) cohomologytheories In sect72 making use of the noncommutative realization associated to deRham-Betti cohomology we extend Grothendieckrsquos theory of periods to the broadnoncommutative setting of dg categories As an application we prove that (modulo2πi) Grothendieckrsquos theory of periods is HPD-invariant

71 Noncommutative realizations Let F be a field of characteristic zero and(Cotimes1) an F -linear neutral Tannakian category equipped with a otimes-invertibleldquoTaterdquo object 1(1) In what follows we write Gal(C) for the Tannakian Galoisgroup of C and Gal0(C) for the kernel of the homomorphism Gal(C) Gm whereGm agrees with the Tannakian Galois group of the smallest Tannakian subcate-gory of C containing 1(1) As explained in [83 sect1-2] given a cohomology theory

Hlowast Sm(k)rarr GrbZ(C) we can consider the associated modified cohomology theory

Hlowast2 Sm(k) minusrarr RepZ2(Gal0(C)) X 7rarr (

oplusn even

Hn(X)oplusn odd

Hn(X))


with values in the category of finite-dimensional Z2-graded continuous represen-tations of Gal0(C) Examples of cohomology theories include Norirsquos cohomologytheory HlowastN (with values in Norirsquos Tannakian category of mixed motives [33 sect8])Jannsenrsquos cohomology theory HlowastJ (with values in Jannsenrsquos Tannakian category ofmixed motives [37 Part I]) de Rham cohomology theory HlowastdR (with values in theTannakian category of finite-dimensional k-vector spaces) Betti cohomology the-ory HlowastB (with values in the Tannakian category finite-dimensional Q-vector spaces)de Rham-Betti cohomology theory HlowastdRB (with values in the Tannakian categoryVect(kQ) of triples (VW ω) where V is a finite-dimensional k-vector space W isa finite-dimensional Q-vector space and ω is an isomorphism V otimesk C W otimesQ C)etale l-adic cohomology theory Hlowastl-adic (with values in the Tannakian category offinite-dimensional l-adic representations of the absolute Galois group of k) Hodgecohomology theory HlowastHod (with values in the Tannakian category of mixed Q-Hodgestructures [77 sect1]) etc consult [83 sect2] for details and for further examples Eachone of these cohomology theories gives rise to a modified cohomology theory

The (proof of the) next result is contained in [83 Thms 12 22 and Prop 31]

Theorem 71 Given a cohomology theory Hlowast there exists an additive invariant

(72) Hlowast2nc dgcat(k) minusrarr Ind(RepZ2(Gal0(C)))

with values in the category of ind-objects such that Hlowast2nc(perfdg(X)) H

lowast2 (X) for

every smooth k-scheme X

The additive invariant (72) is called the noncommutative realization associatedto the cohomology theory Hlowast Morally speaking Theorem 71 shows that themodified cohomology theories belong not only to the realm of algebraic geometrybut also to the broad noncommutative setting of dg categories This insight goesback to Kontsevichrsquos definition of noncommutative etale cohomology theory see[52] Among other ingredients the proof of Theorem 71 makes use of Theorem 67

Remark 73 (Generalizations) (i) In the case where k is of characteristic zeroSm(k) can be replaced by the category of k-schemes

(ii) By construction (72) can be promoted to a localizing invariant

The following result describes the behavior of the noncommutative realizationswith respect to sheaves of differential operators in characteristic zero11

Theorem 74 ([83 Thm 34]) Let k be a field of characteristic zero X a smoothk-scheme and DX the sheaf of differential operators on X Given a cohomology

theory Hlowast we have an isomorphism Hlowast2nc(perfdg(DX)) H lowast

2 (X)

Example 75 (Lie algebras) Let G be a connected semisimple algebraic C-groupB a Borel subgroup of G g the Lie algebra of G and Uev(g)I the quotient of theuniversal enveloping algebra of g by the kernel of the trivial character Thanks toBeilinson-Bernsteinrsquos celebrated ldquolocalisationrdquo theorem [6] it follows from Theorem

74 that Hlowast2nc(Uev(g)I) H

lowast2nc(perfdg(DGB)) H lowast

2 (GB)

Remark 76 Theorem 74 does not holds for every additive invariant For examplein the case of Hochschild homology we have HHlowast(perfdg(DX)) H2dminuslowast

dR (X) for

11Consult [81 Example 220] for the description of the behavior of all additive invariants withrespect to sheaves of differential operators in positive characteristic

26 GONCALO TABUADA

every smooth affine k-scheme X of dimension d see Wodzicki [111 Thm 2] SinceH2ddR(X) = 0 this implies that HH(perfdg(DX)) 6 HH(X) More generally we

have HH(perfdg(DX)) 6 HH(A) for every commutative k-algebra A

72 Periods In this subsection we assume that the base field is endowed withan embedding k rarr C Consider the Z-graded C-algebra of Laurent polynomialsC[t tminus1] with t of degree 1 Given a triple (VW ω) isin Vect(kQ) let us writeP(VW ω) sube C for the subset of entries of the matrix representations of ω (withrespect to basis of V and W ) In the same vein given an object (VnWn ωn)nisinZof the category GrbZ(Vect(kQ)) let us write P((VnWn ωn)nisinZ) for the Z-graded k-subalgebra of C[t tminus1] generated in degree n by the elements of the setP (WnWn ωn) In the case of a smooth k-scheme X P(X) = P(HlowastdRB(X)) iscalled the (Z-graded) algebra of periods of X This algebra originally introducedby Grothendieck in the sixties plays a key role in the study of transcendentalnumbers consult for example the work of Kontsevich-Zagier [54]

Consider the quotient homomorphism φ C[t tminus1] C[t tminus1]〈1minus(2πi)t2〉 Thenext result extends Grothendieckrsquos theory of periods from schemes to dg categories

Theorem 77 ([83 Thm 41]) There exists an assignment A 7rarr Pnc(A) withPnc(A) a Z2-graded k-subalgebra of C[t tminus1]〈1minus(2πi)t2〉 such that Pnc(perfdg(X))is isomorphic to φ(P(X)) for every smooth k-scheme X

Morally speaking Theorem 77 shows that Grothendieckrsquos theory of periods canbe extended from schemes to the broad noncommutative setting of dg categories aslong as we work modulo 2πi Among other ingredients its proof makes use of thenoncommutative realization associated to de Rham-Betti cohomology theory

721 Homological projective duality Let X and Y be two HP-dual smooth pro-jective k-schemes as in sect321 Recall from loc cit that the category perf(X)admits in particular a Lefschetz decomposition 〈A0A1(1) Aiminus1(iminus1)〉 Givena linear subspace L sub W lowast consider the linear sections XL = X timesP(Wlowast) P(L) and

YL = Y timesP(W )P(Lperp) The next result proved in [83 Thm 46] relates the algebraof periods of XL with the algebra of periods of YL

Theorem 78 (HPD-invariance) Let X and Y be as above Assume that XL

and YL are smooth that dim(XL) = dim(X)minus dim(L) that dim(YL) = dim(Y )minusdim(Lperp) and that the category A0 admits a full exceptional collection Under theseassumptions the Z2-graded k-algebras φ(P(XL)) and φ(P(YL)) are isomorphic

Roughly speaking Theorem 78 shows that (modulo 2πi) Grothendieckrsquos theoryof periods is invariant under homological projective duality This result can beapplied for example to the Veronese-Clifford duality to the Spinor duality to theGrassmannian-Pfaffian duality to the Determinantal duality etc

References

[1] Y Andre Une introduction aux motifs (motifs purs motifs mixtes periodes) Panoramas etSyntheses vol 17 Societe Mathematique de France Paris 2004

[2] P Ara M Barroso K Goodearl and E Pardo Fractional skew monoid rings J Algebra278 (2004) no 1 104ndash126

[3] M Artin and J Zhang Noncommutative projective schemes Adv Math 109 (1994) no 2

228ndash287[4] V Baranovsky Orbifold cohomology as periodic cyclic homology Internat J Math 14

(2003) no 8 791ndash812


[5] H Bass Some problems in classical algebraic K-theory Algebraic K- theory II ldquoClassicalrdquo

algebraic K-theory and connections with arithmetic (Proc Conf Battelle Memorial Inst

Seattle Wash 1972) pp 3ndash73 LNM 342 1973[6] A Beilinson and J Bernstein Localisation de g-modules C R Acad Sci Paris Ser I Math

292 (1981) no 1 15ndash18

[7] M Bernardara M Marcolli and G Tabuada Some remarks concerning Voevodskyrsquos nilpo-tence conjecture Available at arXiv14030876 To appear in J Reine Angew Math

[8] R Bezrukavnikov and D Kaledin McKay equivalence for symplectic resolutions of quotient

singularities Tr Mat Inst Steklova 246 (2004) Alg Geom Metody Svyazi i Prilozh 20ndash42[9] A Bialynicki-Birula Some theorems on actions of algebraic groups Ann of Math (2) 98

(1973) 480ndash497

[10] S Bloch A Kas and D Lieberman Zero cycles on surfaces with pg = 0 Compositio Math33 (1976) 135ndash145

[11] A I Bondal and M M Kapranov Representable functors Serre functors and reconstruc-tions Izv Akad Nauk SSSR Ser Mat 53 (1989) no 6 1183ndash1205 1337

[12] A I Bondal M Larsen and V A Lunts Grothendieck ring of pretriangulated categories

Int Math Res Not (2004) no 29 1461ndash1495[13] A I Bondal and D O Orlov Semiorthogonal decomposition for algebraic varieties Available

at arXivalg-geom9506012

[14] M Bondarko and G Tabuada Picard groups weight structures and (noncommutative)mixed motives Available at arXiv151209101 To appear in Doc Math

[15] T Bridgeland A King and M Reid The McKay correspondence as an equivalence of

derived categories J Amer Math Soc 14 (2001) no 3 535ndash554[16] V Chernousov S Gille and A Merkurjev Motivic decomposition of isotropic projective

homogeneous varieties Duke Math J 126 no 1 (2005) 137ndash159

[17] P Deligne Categories tensorielles Mosc Math J 2 (2002) no 2 227ndash248 Dedicated toYuri I Manin on the occasion of his 65th birthday

[18] La conjecture de Weil I Inst Hautes Etudes Sci Publ Math 43 (1974) 273ndash307[19] P Deligne and J S Milne Tannakian categories In Hodge Cycles Motives and Shimura

Varieties Lecture Notes in Mathematics vol 900 pp 101ndash228 Updated online version

available at httpwwwjmilneorgmathxnotestcpdf (1982)[20] B Dwork On the rationality of the zeta function of an algebraic variety Amer J Math 82

(1960) 631ndash648[21] D Edidin and W Graham Equivariant intersection theory Inv Math 131 (1998) 595ndash634

[22] B Feigin and A Odesskii Sklyaninrsquos elliptic algebras Funct Anal Appl 23 (1990) no 3

207ndash214[23] O Gabber Some theorems on Azumaya algebras The Brauer group (Sem Les Plans-sur-

Bex 1980) 129ndash209

[24] P Gabriel Des categories abeliennes Bull Soc Math France 90 (1962) 323ndash448[25] A Grothendieck Standard conjectures on algebraic cycles 1969 Algebraic Geometry (Inter-

nat Colloq Tata Inst Fund Res Bombay 1968) pp 193ndash199 Oxford Univ Press London

[26] A Grothendieck Formule de Lefschetz et rationalite des fonctions L Seminaire Bourbaki279 (1965)

[27] V Guletskii Finite-dimensional objects in distinguished triangles J Number Theory 119

(2006) no 1 99ndash127[28] V Guletskii and C Pedrini Finite-dimensional motives and the conjectures of Beilinson and

Murre Special issue in honor of Hyman Bass on his seventieth birthday Part III K-Theory30 (2003) no 3 243ndash263

[29] D Grayson Finite generation of K-groups of a curve over a finite field (after DanielQuillen) Algebraic K-theory Part I (Oberwolfach 1980) pp 69ndash90 LNM 966 1982

[30] C Haesemeyer and J Hornbostel Motives and etale motives with finite coefficients K-Theory34 (2005) no 3 195ndash207

[31] R Hartshorne On the de Rham cohomology of algebraic varieties Inst Hautes Etudes Sci

Publ Math (1971) no 45 5ndash99[32] M Hazewinkel Witt vectors I Handbook of algebra Vol 6 319ndash472 North-Holland Ams-

terdam 2009

[33] A Huber and S Muller-Stach Periods and Nori motives Book available at the webpagehttphomemathematikuni-freiburgdearithgeomforschunghtml

28 GONCALO TABUADA

[34] F Ivorra Levinersquos motivic comparison theorem revisited J Reine Angew Math 617 (2008)

67ndash107

[35] J Iyer and S Muller-Stach Chow-Kunneth decomposition for some moduli spaces DocMath 14 (2009) 1ndash18

[36] U Jannsen Motives numerical equivalence and semi-simplicity Invent Math 107 (1992)

no 3 447ndash452[37] Mixed motives and algebraic K-theory With appendices by S Bloch and C Schoen

LNM 1400 Springer-Verlag Berlin 1990

[38] A de Jong A result of Gabber Available online at the personal webpage of Aise Johan deJong httpwwwmathcolumbiaedudejongpapers2-gabberpdf

[39] B Kahn Zeta functions and motives Pure Appl Math Quarterly 5 (2009) 507ndash570

[40] Motifs et adjoints Available at arXiv150608386

[41] B Kahn and R Sebastian Smash-nilpotent cycles on abelian 3-folds Math Res Lett 16

(2009) no 6 1007ndash1010[42] M Kapranov and E Vasserot Kleinian singularities derived categories and Hall algebras

Math Ann 316 (2000) no 3 565ndash576

[43] N Karpenko Cohomology of relative cellular spaces and of isotropic flag varieties Algebrai Analiz 12 (2000) no 1 3ndash69

[44] Y Kawamata D-equivalence and K-equivalence J Differential Geom 61(1) (2002) 147ndash171

[45] B Keller On differential graded categories International Congress of Mathematicians VolII Eur Math Soc Zurich 2006 pp 151ndash190

[46] S-I Kimura Chow groups are finite dimensional in some sense Math Ann 331 (2005)

no 1 173ndash201[47] S L Kleiman The standard conjectures Motives (Seattle WA 1991) 3ndash20 Proc Sympos

Pure Math 55 Part 1 Amer Math Soc Providence RI 1994

[48] Algebraic cycles and the Weil conjectures Dix exposes sur la cohomologie des

schemas 359ndash386 Adv Stud Pure Math 3 North-Holland Amsterdam 1968

[49] J Kollar Conics in the Grothendieck ring Adv Math 198 (2005) no 1 27ndash35[50] Maxim Kontsevich Mixed noncommutative motives Talk at the Workshop on Homological

Mirror Symmetry Miami 2010 Available at www-mathmiteduaurouxfrgmiami10-notes

[51] Notes on motives in finite characteristic Algebra arithmetic and geometry inhonor of Yu I Manin Vol II Progr Math vol 270 Birkhauser Boston Inc Boston MA

2009 pp 213ndash247

[52] Categorification NC Motives Geometric Langlands and Lattice Models Talk at theGeometric Langlands Seminar University of Chicago 2006 Notes available at the webpage

httpswwwmautexaseduusersbenzvinoteshtml

[53] Noncommutative motives Talk at the IAS on the occasion of the 61st birthday of

Pierre Deligne (2005) Available at httpvideoiaseduGeometry-and-Arithmetic

[54] M Kontsevich and D Zagier Periods Mathematics unlimitedndash2001 and beyond 771ndash808Springer Berlin 2001

[55] A Kuznetsov Semiorthogonal decompositions in algebraic geometry Available at 14043143

To appear in Proceedings of the ICM 2014

[56] Homological projective duality Inst Hautes Etudes Sci Publ Math (2007) no 105157ndash220

[57] Hyperplane sections and derived categories Izv Ross Akad Nauk Ser Mat 70

(2006) no 3 23ndash128 translation in Izv Math 70 (2006) no 3 447ndash547

[58] Homological projective duality for Grassmannians of lines Available atarXivmath0610957

[59] R Laterveer Equivariant motives Indag Math (NS) 9 (1998) no 2 255ndash275

[60] W Leavitt The module type of a ring Trans Amer Math Soc 103 113ndash130 (1962)[61] M Levine Mixed motives Mathematical Surveys and Monographs vol 57 American Math-

ematical Society Providence RI 1998

[62] D Lieberman Numerical and homological equivalence of algebraic cycles on Hodge manifoldsAmer J Math 90 366ndash374 1968

[63] Y Manin Quantum groups and noncommutative geometry Universite de Montreal Centre

de Recherches Mathematiques Montreal QC 1988[64] Some remarks on Koszul algebras and quantum groups Ann Inst Fourier 37 (1987)

no 4 191ndash205


[65] C Mazza Schur functors and motives K-Theory 33 (2004) no 2 89ndash106

[66] F Morel and V Voevodsky A1-homotopy theory of schemes Inst Hautes Etudes Sci PublMath (1999) no 90 45ndash143 (2001)

[67] D Mumford The topology of normal singularities of an algebraic surface and a criterion for

simplicity Inst Hautes Etudes Sci Publ Math No 9 (1961) 5ndash22

[68] I Panin On the algebraic K-theory of twisted flag varieties K-Theory 8 (1994) no 6

541ndash585[69] D Quillen Finite generation of the groups Ki of rings of algebraic integers Cohomology of

groups and algebraic K-theory 479ndash488 Adv Lect Math (ALM) 12 (2010)[70] Higher algebraic K-theory I Algebraic K-theory I Higher K-theories (Proc Conf

Battelle Memorial Inst Seattle Wash 1972) Springer Berlin 1973 pp 85ndash147 Lecture

Notes in Math Vol 341[71] On the cohomology and K-theory of the general linear groups over a finite field

Ann of Math (2) 96 (1972) 552ndash586

[72] M Reid The complete intersection of two or more quadrics Ph D thesis Available at thewebpage httpshomepageswarwickacuk~masda3foldsqupdf

[73] J-P Serre Faisceaux algebriques coherents Ann of Math 61(2) (1955) 197ndash278

[74] A Shermenev The motive of an abelian variety Funct Anal 8 (1974) 47ndash53[75] S P Smith Some finite-dimensional algebras related to elliptic curves Representation theory

of algebras and related topics (Mexico City 1994) 315ndash348 CMS Conf Proc 19 Amer

Math Soc Providence RI 1996[76] J T Stafford Noncommutative projective geometry Proceedings of the International Con-

gress of Mathematicians vol II Beijing 2002 Higher Ed Press Beijing 2002 pp 93ndash103[77] J Steenbrink A summary of mixed Hodge theory Motives I Proc Symp Pure Math Vol

55 AMS 1991 pp 31ndash42

[78] D Stephenson and J Zhang Noetherian connected graded algebras of global dimension 3 JAlgebra 230 (2000) no 2 474ndash495

[79] A Suslin On the K-theory of algebraically closed fields Invent Math 73 (1983) no 2

241ndash245[80] G Tabuada A note on secondary K-theory Algebra and Number Theory 10 (2016) no 4

887ndash906

[81] Noncommutative motives With a preface by Yuri I Manin University Lecture Series63 American Mathematical Society Providence RI 2015

[82] Equivariant noncommutative motives Available at arXiv151105501 To appear inAnnals of K-theory

[83] Modified mixed realizations new additive invariants and periods of dg categories

Available at arXiv160303411 To appear in Int Math Res Not[84] A note on Grothendieckrsquos standard conjectures of type C+ and D Available at

arXiv160505307 To appear in Proceedings of the AMS

[85] A note on the Schur-finiteness of linear sections Available at arXiv161006553 Toappear in Math Res Lett

[86] A1-homotopy invariants of corner skew Laurent polynomial algebras Available at

arXiv160309737 To appear in Journal of Noncommutative Geometry[87] Jacques Titsrsquo motivic measure Available at arXiv160406407

[88] A note on secondary K-theory II Available at arXiv160703094

[89] Invariants of noncommutative projective schemes Available at arXiv170204712

[90] Noncommutative rigidity Available at arXiv170310599

[91] HPD-invariance of the Tate conjecture Available at arXiv170706639[92] Bass and Schur finiteness conjectures for quadric fibrations Available at

arXiv170805382

[93] Noncommutative motives in positive characteristic and their applications Availableat arXiv170704248

[94] G Tabuada and M Van den Bergh Noncommutative motives of separable algebras Adv

Math 303 (2016) 1122ndash1161[95] The Gysin triangle via localization and A1-homotopy invariance Available at

arXiv151004677 To appear in Transactions of the AMS

[96] Additive invariants of orbifolds Available at arXiv161203162

30 GONCALO TABUADA

[97] J Tate Conjectures on algebraic cycles in l-adic cohomology Motives (Seattle WA 1991)

71ndash83 Proc Sympos Pure Math 55 Part 1 Amer Math Soc Providence RI 1994

[98] Algebraic cycles and poles of zeta functions Arithmetical Algebraic Geometry (ProcConf Purdue Univ 1963) pp 93ndash110 Harper amp Row New York 1965

[99] J Tate and M Van den Bergh Homological properties of Sklyanin algebras Invent Math

124 (1996) no 1-3 619ndash647[100] R W Thomason and T Trobaugh Higher algebraic K-theory of schemes and of derived

categories The Grothendieck Festschrift Vol III Progr Math vol 88 Birkhauser Boston

Boston MA 1990 pp 247ndash435[101] B Toen Secondary K-theory Talk at the Workshop on Topological Field Theories North-

western University 2009 Available online at David Ben-Zvirsquos personal webpage

httpswwwmautexaseduusersbenzviGRASPlecturesNWTFTtoennwpdf[102] B Totaro Recent progress on the Tate conjecture Available at Burt Totarorsquos personal

webpage httpwwwmathuclaedu~totaropaperspublic_htmlindexhtml[103] A Vistoli Higher equivariant K-theory for finite group actions Duke Math J 63 (1991)

no 2 399ndash419

[104] V Voevodsky Triangulated categories of motives over a field Cycles transfers and motivichomology theories Ann of Math Stud vol 143 Princeton Univ Press 2000 pp 188ndash238

[105] A1-homotopy theory Proceedings of the International Congress of Mathematicians

Vol I (Berlin 1998) no Extra Vol I 1998 pp 579ndash604 (electronic)[106] A nilpotence theorem for cycles algebraically equivalent to zero Int Math Res

Not IMRN 1995 (1995) no 4 187ndash198

[107] C Voisin Blochrsquos conjecture for Catanese and Barlow surfaces J Differential Geom 97(2014) no 1 149ndash175

[108] Remarks on zero-cycles of self-products of varieties in Moduli of vector bundles(Sanda 1994 Kyoto 1994) Lecture Notes in Pure and Appl Math 179 Dekker New York

(1996) 265ndash285

[109] Sur les zero-cycles de certaines hypersurfaces munies drsquoun automorphisme AnnScuola Norm Sup Pisa 19 (1992) 473ndash492

[110] A Weil Varietes abeliennes et courbes algebriques Hermann Paris (1948)

[111] M Wodzicki Cyclic homology of differential operators Duke Math J 54 (1987) no 2641ndash647

Goncalo Tabuada Department of Mathematics MIT Cambridge MA 02139 USAE-mail address tabuadamathmitedu

URL httpmathmitedu~tabuada

Introduction








References

2 GONCALO TABUADA

In this survey we cover some of the recent developments concerning the Chapters2 4 5 6 8 and 9 These developments are described in Sections 1 2 3 4 5 and6 respectively The final Section 7 entitled ldquoNoncommutative realizations andperiodsrdquo discusses a recent research subject which was not addressed in [81]

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 Kellerrsquos ICM address [45] Inparticular we will freely use the notions of Morita equivalence of dg categories (see[81 sect16]) and smoothproper dg category in the sense of Kontsevich (see [81 sect17])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 [81 Example 127]


Recall from [81 sect23] the construction of the universal additive invariant of dgcategories U dgcat(k) rarr Hmo0(k) In [81 sect24] we described the behavior of Uwith respect to semi-orthogonal decompositions full exceptional collections purelyinseparable field extensions central simple algebras sheaves of Azumaya algebrastwisted flag varieties nilpotent ideals finite-dimensional algebras of finite globaldimension etc In sect11-12 we describe the behavior of U with respect to relativecellular spaces and orbifolds As explained in [81 Thm 29] all the results in sect11-12 are motivic in the sense that they hold similarly for every additive invariant suchas algebraic K-theory mod-n algebraic K-theory Karoubi-Villamayor K-theorynonconnective algebraic K-theory homotopy K-theory etale K-theory Hochschildhomology cyclic homology negative cyclic homology periodic cyclic homologytopological Hochschild homology topological cyclic homology topological periodiccyclic homology etc Consult sect71 for further examples of additive invariants


11 Relative cellular spaces A flat morphism of k-schemes p X rarr Y is calledan affine fibration of relative dimension d if for every point y isin Y there existsa Zariski open neighborhood y isin V such that XV = pminus1(V ) Y times Ad withpV XV rarr Y isomorphic to the projection onto the first factor Following Karpenko[43 Def 61] a smooth projective k-scheme X is called a relative cellular space ifit admits a filtration by closed subschemes

empty = Xminus1 rarr X0 rarr middot middot middot rarr Xi rarr middot middot middot rarr Xnminus1 rarr Xn = X

and affine fibrations pi XiXiminus1 rarr Yi 0 le i le n of relative dimension di with Yia smooth projective k-scheme The smooth k-schemes XiXiminus1 are called the cellsand the smooth projective k-schemes Yi the bases of the cells

Example 12 (Gm-schemes) The celebrated Bialynicki-Birula decomposition [9]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 [43] as well as the isotropichomogeneous spaces considered by Chernousov-Gille-Merkurjev in [16]



oplusni=0 U(Yi)
















U(Xg)C(g)F (

oplusgisinG

U(Xg)F )G



4 GONCALO TABUADA


(110) U(Km(S))12 U(S)C2









HClowast(Xg)C(g) (

oplusgisinG

HClowast(Xg))G





oplusgisinG

TPlowast(Xg)1p)

G


















(23) [A] = [B]hArr U(A) U(B)


6 GONCALO TABUADA


(25) U(A) U(B)hArr [U(A)] [U(B)]














et(XGm)
















(216) Z(f t) = exp

sumnge1

tr(fn)tn

n

isin QJtK



tm

m isin QJtK

8 GONCALO TABUADA




)] isin K0(k) Z





A B



n

n ) of X







δidi




















0 (Aop otimes B)







hG(minus)Q

dgcatGsp(k)

UG(minus)Q

ChowG(k)Q

NChowG(k)Q

(minus)IQ


NChowG(k)QIQ



10 GONCALO TABUADA
















X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References



oplusni=0 U(Yi)
















U(Xg)C(g)F (

oplusgisinG

U(Xg)F )G



4 GONCALO TABUADA


(110) U(Km(S))12 U(S)C2









HClowast(Xg)C(g) (

oplusgisinG

HClowast(Xg))G





oplusgisinG

TPlowast(Xg)1p)

G


















(23) [A] = [B]hArr U(A) U(B)


6 GONCALO TABUADA


(25) U(A) U(B)hArr [U(A)] [U(B)]














et(XGm)
















(216) Z(f t) = exp

sumnge1

tr(fn)tn

n

isin QJtK



tm

m isin QJtK

8 GONCALO TABUADA




)] isin K0(k) Z





A B



n

n ) of X







δidi




















0 (Aop otimes B)







hG(minus)Q

dgcatGsp(k)

UG(minus)Q

ChowG(k)Q

NChowG(k)Q

(minus)IQ


NChowG(k)QIQ



10 GONCALO TABUADA
















X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

4 GONCALO TABUADA


(110) U(Km(S))12 U(S)C2









HClowast(Xg)C(g) (

oplusgisinG

HClowast(Xg))G





oplusgisinG

TPlowast(Xg)1p)

G


















(23) [A] = [B]hArr U(A) U(B)


6 GONCALO TABUADA


(25) U(A) U(B)hArr [U(A)] [U(B)]














et(XGm)
















(216) Z(f t) = exp

sumnge1

tr(fn)tn

n

isin QJtK



tm

m isin QJtK

8 GONCALO TABUADA




)] isin K0(k) Z





A B



n

n ) of X







δidi




















0 (Aop otimes B)







hG(minus)Q

dgcatGsp(k)

UG(minus)Q

ChowG(k)Q

NChowG(k)Q

(minus)IQ


NChowG(k)QIQ



10 GONCALO TABUADA
















X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References
















(23) [A] = [B]hArr U(A) U(B)


6 GONCALO TABUADA


(25) U(A) U(B)hArr [U(A)] [U(B)]














et(XGm)
















(216) Z(f t) = exp

sumnge1

tr(fn)tn

n

isin QJtK



tm

m isin QJtK

8 GONCALO TABUADA




)] isin K0(k) Z





A B



n

n ) of X







δidi




















0 (Aop otimes B)







hG(minus)Q

dgcatGsp(k)

UG(minus)Q

ChowG(k)Q

NChowG(k)Q

(minus)IQ


NChowG(k)QIQ



10 GONCALO TABUADA
















X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

6 GONCALO TABUADA


(25) U(A) U(B)hArr [U(A)] [U(B)]














et(XGm)
















(216) Z(f t) = exp

sumnge1

tr(fn)tn

n

isin QJtK



tm

m isin QJtK

8 GONCALO TABUADA




)] isin K0(k) Z





A B



n

n ) of X







δidi




















0 (Aop otimes B)







hG(minus)Q

dgcatGsp(k)

UG(minus)Q

ChowG(k)Q

NChowG(k)Q

(minus)IQ


NChowG(k)QIQ



10 GONCALO TABUADA
















X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References
















(216) Z(f t) = exp

sumnge1

tr(fn)tn

n

isin QJtK



tm

m isin QJtK

8 GONCALO TABUADA




)] isin K0(k) Z





A B



n

n ) of X







δidi




















0 (Aop otimes B)







hG(minus)Q

dgcatGsp(k)

UG(minus)Q

ChowG(k)Q

NChowG(k)Q

(minus)IQ


NChowG(k)QIQ



10 GONCALO TABUADA
















X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

8 GONCALO TABUADA




)] isin K0(k) Z





A B



n

n ) of X







δidi




















0 (Aop otimes B)







hG(minus)Q

dgcatGsp(k)

UG(minus)Q

ChowG(k)Q

NChowG(k)Q

(minus)IQ


NChowG(k)QIQ



10 GONCALO TABUADA
















X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References











0 (Aop otimes B)







hG(minus)Q

dgcatGsp(k)

UG(minus)Q

ChowG(k)Q

NChowG(k)Q

(minus)IQ


NChowG(k)QIQ



10 GONCALO TABUADA
















X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

10 GONCALO TABUADA
















X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References



X =sumn π

2nX is algebraic










Gal(kk)








A) = π+A






X =sum



12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

12 GONCALO TABUADA



)n ge 1 (33)































14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References














14 GONCALO TABUADA




Gm)













notation UA1






















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References



















16 GONCALO TABUADA















oplusn evenH

ndR(X)

oplusn oddH







n evenHndR(Z)

oplusnH

ndR(ilowast) oplus

n evenHndR(X)

oplusnH

ndR(jlowast) oplus

n evenHndR(V )

part

oplusn oddH

ndR(V )

part

OO

oplusn oddH

ndR(X)oplus

nHndR(jlowast)

oo oplusn oddH

ndR(Z) oplus

nHndR(ilowast)

oo












n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References











n+ with










18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

18 GONCALO TABUADA





sumiisinQ0



























d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References






d of degree d







x2x3 = x3x2 x2x4 = qx4x2 x1x4 = qx4x3










20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

20 GONCALO TABUADA


(525)


U(k)Mminusrarr




U(k)[1]
















(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References





(64) Sm(k)

Σinfin(minus+)

X 7rarrperfdg(X)

))

dgcat(k)

U

SH(k)

minusandKGL

NMix(k)

(minus)or

NMot(k)

Hom(minusU(k))

Mod(KGL)





Sm(k)

Σinfin(minus+)1p

X 7rarrperfdg(X)

))

dgcat(k)

U(minus)1p

SH(k)1p

minusandKGL1p

NMix(k)1p

(minus)or

NMot(k)1p

Hom(minusU(k)1p)

Mod(KGL1p) Ψ1p






22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

22 GONCALO TABUADA





(68) Sm(k)

M(minus)Q

X 7rarrperfdg(X)

dgcat(k)

U(minus)Q

DMgm(k)Q

NMix(k)Q

(minus)or

NMot(k)Q

Hom(minusU(k)Q)


NMix(k)Q NMot(k)Q













(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References



(612) Sm(k)

h(minus)Q


U(minus)Q

DM(k)Q


NMix(k)Q NMot(k)Q












24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

24 GONCALO TABUADA














oplusn even

Hn(X)oplusn odd

Hn(X))







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References







lowast2 (X) for








2 (X)




2 (GB)


dR (X) for


26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

26 GONCALO TABUADA












References





(2003) no 8 791ndash812





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References





292 (1981) no 1 15ndash18




(1973) 480ndash497

















Bex 1980) 129ndash209











terdam 2009


28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

28 GONCALO TABUADA


67ndash107










Math Ann 316 (2000) no 3 565ndash576












2009 pp 213ndash247


















no 4 191ndash205











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References











Ann of Math (2) 96 (1972) 552ndash586











887ndash906













arXiv170805382






30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

30 GONCALO TABUADA











no 2 399ndash419




Not IMRN 1995 (1995) no 4 187ndash198



(1996) 265ndash285






Introduction








References

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