Table Of ContentUniversity
L
ECTURE
Series
Volume 63
Noncommutative Motives
Gonçalo Tabuada
American Mathematical Society
Noncommutative Motives
University
L
ECTURE
Series
Volume 63
Noncommutative Motives
Gonçalo Tabuada
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Jordan S. Ellenberg Robert Guralnik
WilliamP. Minicozzi II (Chair) Tatiana Toro
2010 Mathematics Subject Classification. Primary 14A22, 14C15, 18D20;
Secondary 18E30, 18G55, 19D55.
For additional informationand updates on this book, visit
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Tabuada,Gon¸calo,1979–
Noncommutativemotives/Gon¸caloTabuada.
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Includesbibliographicalreferencesandindex.
ISBN978-1-4704-2397-1(alk.paper)
1.Motives(Mathematics) 2.Noncommutativealgebras. 3.Algebraicvarieties. I.Title.
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To Lily, for not being by my side.
Contents
Preface ix
Introduction 1
Chapter 1. Differential graded categories 3
1.1. Definitions 3
1.2. Quasi-equivalences 5
1.3. Drinfeld’s DG quotient 11
1.4. Pretriangulated equivalences 13
1.5. Bondal-Kapranov’s pretriangulated envelope 14
1.6. Morita equivalences 15
1.7. Kontsevich’s smooth proper dg categories 16
Chapter 2. Additive invariants 21
2.1. Definitions 21
2.2. Examples 22
2.3. Universal additive invariant 26
2.4. Computations 29
2.5. Lefschetz’s fixed point formula 32
Chapter 3. Background on pure motives 35
Chapter 4. Noncommutative pure motives 41
4.1. Noncommutative Chow motives 41
4.2. Relation with Chow motives 41
4.3. Relation with Merkurjev-Panin’s motives 46
4.4. Noncommutative ⊗-nilpotent motives 47
4.5. Noncommutative homological motives 47
4.6. Noncommutative numerical motives 48
4.7. Kontsevich’s noncommutative numerical motives 49
4.8. Semi-simplicity 50
4.9. Noncommutative Artin motives 52
4.10. Functoriality 53
4.11. Weil restriction 54
Chapter 5. Noncommutative (standard) conjectures 57
5.1. Standard conjecture of type C 57
nc
5.2. Standard conjecture of type D 58
nc
5.3. Noncommutative nilpotence conjecture 59
5.4. Kimura-finiteness 59
5.5. All together 60
vii
viii CONTENTS
Chapter 6. Noncommutative motivic Galois groups 63
6.1. Definitions 63
6.2. Relation with motivic Galois groups 65
6.3. Unconditional version 65
6.4. Base-change short exact sequence 66
Chapter 7. Jacobians of noncommutative Chow motives 69
Chapter 8. Localizing invariants 71
8.1. Definitions 71
8.2. Examples 72
8.3. Universal localizing invariant 73
8.4. Additivity 77
8.5. A1-homotopy 79
8.6. Algebraic K-theory 82
8.7. Witt vectors 84
8.8. Natural transformations 85
Chapter 9. Noncommutative mixed motives 87
9.1. Definitions 87
9.2. Relation with noncommutative Chow motives 89
9.3. Weight structure 89
9.4. Relation with Morel-Voevodsky’s motivic homotopy theory 90
9.5. Relation with Voevodsky’s geometric mixed motives 91
9.6. Relation with Levine’s mixed motives 93
9.7. Noncommutative mixed Artin motives 93
9.8. Kimura-finiteness 94
9.9. Coefficients 95
Chapter 10. Noncommutative motivic Hopf dg algebras 97
10.1. Definitions 97
10.2. Base-change short exact sequence 98
Appendix A. Grothendieck derivators 99
A.1. Definitions 99
A.2. Left Bousfield localization 101
A.3. Stabilization and spectral enrichment 102
A.4. Filtered homotopy colimits 102
A.5. Symmetric monoidal structures 103
Bibliography 105
Index 113
Preface
Alexandre Grothendieck conceived his definition of motives in the 1960s. By
that time, it was already established that there exist several cohomology theories
for, say, smooth projective algebraic varieties defined over a given field k, and
A. Weil’s brilliant insight about counting points over finite fields via the Lefschetz
trace formula was validated.
With his characteristic passion for unification and “naturality”, Grothendieck
wanted to construct a universal cohomology theory (with, say, coefficients R) that
had to be a functor h from the category Var(k) of smooth k-varieties to an abelian
tensor category Mot(k) of “(pure) motives” (or Mot(k) , where R is a ring of
R
coefficients), satisfying a minimal list of expected properties.
Grothendieck also suggested a definition of Mot(k) and of the motivic functor.
It consisted of several steps.
For the first step, one keeps objects of Var(k), but replaces its morphisms
by correspondences. This passage implies that morphisms Y → X now form an
additive group, or even an R-module rather than simply a set. Moreover, cor-
respondences themselves are not just cycles on X ×Y but classes of such cycles
modulo an “adequate” equivalence relation. The coarsest such relation is that of
numerical equivalence, when two equidimensional cycles are equivalent if their in-
tersectionindices witheachcycleofcomplementary dimension coincide. The finest
oneistherational(Chow)equivalence,whenequivalentcyclesarefibresofafamily
parametrized by a chain of rational curves. The direct product of varieties induces
the tensor product structure on the category.
The second step in the definition of the relevant category of pure motives
consists in a formal construction of new objects (and relevant morphisms) that
are “pieces” of varieties: kernels and images of projectors, i.e., correspondences
p:X →X with p2 =p. This produces a pseudo-abelian, or Karoubian completion
ofthecategory. Inthisnewcategory,theprojectivelineP1 becomesthedirectsum
of (motive of) a point and the Lefschetz motive L (intuitively corresponding to the
affine line).
The third, and last step of the construction, is one more formal enhancement
of the class of objects: they now include all integer tensor powers L⊗n, not just
non-negative ones, and tensor products of these with other motives. An important
role is played by L−1 which is called the Tate motive T.
The first twenty-five years of the development of the theory of motives were
summarised in the informative Proceedings of the 1991 Research Conference con-
ference “Motives”, published in two volumes by the AMS in 1994.
By that time it was already clear that the richness of ideas and problems
involved in this projectresists any simple-minded notion of “unification”, and with
time, the theory of motives was more and more resembling a Borgesian garden
ix
