To Lily, for not being by my side.
Contents
Preface
vii
Introduction
1
Chapter 1. Differential graded categories
1.1. Definitions
1.2. Quasi-equivalences
1.3. Drinfeld’s DG quotient
1.4. Pretriangulated equivalences
1.5. Bondal-Kapranov’s pretriangulated envelope
1.6. Morita equivalences
1.7. Kontsevich’s smooth proper dg categories
3
3
5
11
13
14
15
16
Chapter 2. Additive invariants
2.1. Definitions
2.2. Examples
2.3. Universal additive invariant
2.4. Computations
2.5. Lefschetz’s fixed point formula
21
21
22
26
29
32
Chapter 3. Background on pure motives
35
Chapter 4. Noncommutative pure motives
4.1. Noncommutative Chow motives
4.2. Relation with Chow motives
4.3. Relation with Merkurjev-Panin’s motives
4.4. Noncommutative ¢-nilpotent motives
4.5. Noncommutative homological motives
4.6. Noncommutative numerical motives
4.7. Kontsevich’s noncommutative numerical motives
4.8. Semi-simplicity
4.9. Noncommutative Artin motives
4.10. Functoriality
4.11. Weil restriction
41
41
41
46
47
47
48
49
50
52
53
54
Chapter 5. Noncommutative (standard) conjectures
5.1. Standard conjecture of type Cnc
5.2. Standard conjecture of type Dnc
5.3. Noncommutative nilpotence conjecture
5.4. Kimura-finiteness
5.5. All together
57
57
58
59
59
60
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CONTENTS
Chapter 6. Noncommutative motivic Galois groups
6.1. Definitions
6.2. Relation with motivic Galois groups
6.3. Unconditional version
6.4. Base-change short exact sequence
63
63
65
65
66
Chapter 7. Jacobians of noncommutative Chow motives
69
Chapter 8. Localizing invariants
8.1. Definitions
8.2. Examples
8.3. Universal localizing invariant
8.4. Additivity
8.5. A1 -homotopy
8.6. Algebraic K-theory
8.7. Witt vectors
8.8. Natural transformations
71
71
72
73
77
79
82
84
84
Chapter 9. Noncommutative mixed motives
9.1. Definitions
9.2. Relation with noncommutative Chow motives
9.3. Weight structure
9.4. Relation with Morel-Voevodsky’s motivic homotopy theory
9.5. Relation with Voevodsky’s geometric mixed motives
9.6. Relation with Levine’s mixed motives
9.7. Noncommutative mixed Artin motives
9.8. Kimura-finiteness
9.9. Coefficients
87
87
89
89
90
91
93
93
94
95
Chapter 10. Noncommutative motivic Hopf dg algebras
10.1. Definitions
10.2. Base-change short exact sequence
97
97
98
Appendix A. Grothendieck derivators
A.1. Definitions
A.2. Left Bousfield localization
A.3. Stabilization and spectral enrichment
A.4. Filtered homotopy colimits
A.5. Symmetric monoidal structures
99
99
101
102
102
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 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)R , where R is a ring of
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.
At the first step, one keeps objects of Var(k), but replaces its morphisms by correspondences. This passage implies that morphisms Y æ X now form an additive
group, or even a R-module rather than simply a set. Moreover, correspondences
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 intersection indices
with each cycle of complementary dimension coincide. The finest one is the rational
(Chow) equivalence, when equivalent cycles are fibres of a family parametrized by
a chain of rational curves. 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 æ X with p2 = p. This produces a pseudo-abelian, or Karoubian completion
of the category. In this new category, the projective line P1 becomes the direct sum
of the (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 conference “Motives”, published in two volumes by AMS in 1994.
Already by that time it was however clear that the richness of ideas and problems involved in this project resists any simple-minded notion of “unification”, and
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PREFACE
with time, the theory of motives was more and more resembling a Borgesian garden of forking paths. Each strand of the initial project tended to unfold in its own
direction, whereas the central stumbling stone on the Grothendieck visionary road,
the Standard Conjectures, resisted and still resists all efforts.
ııııı
The book by Gonçalo Tabuada is a dense combination of a survey paper and a
research monograph dedicated to the development of the theory of motives during
the next twenty five years. The author contributed many important results and
techniques in the theory in recent years. In this book, he focuses on the so called
“noncommutative motives”. I will make a few brief comments about the scope of
this subject.
In very general terms, one can say that motivic constructions of the New Age
start not with just smooth varieties but rather with triangulated categories and
their enhancements, dg categories. Classical varieties fit there by supplying their
derived and more general enhanced derived categories, such as categories of perfect
complexes. Enhancement essentially means that morphisms rather than objects are
treated as complexes, complexes modulo homotopy, etc. Hence the usual categorical
framework is not sufficient anymore: we must deal with 2-categories and eventually
with categories of higher level.
Correspondences between such “varieties” are introduced using Morita-like
constructions. Recall that in the basic Morita theory morphisms between nonnecessarily commutative rings A æ B are replaced with (A, B)-bimodules, and
that the difference between commutative and noncommutative rings in this framework essentially vanish because any commutative ring is Morita equivalent to the
ring of matrices of any given order over it.
One of the first great surprises of this insight transplanted into (projective)
algebraic geometry was Alexander Beilinson’s discovery (1983) that the derived
category of coherent sheaves of a projective space can be described as a triangulated category made out of modules over a Grassmann algebra. In particular, a
projective space became “affine” in some kind of noncommutative geometry! The
development of Beilinson’s technique led to a general machinery describing triangulated categories in terms of exceptional systems and expanding the realm of
candidates to the role of noncommutative motives.
Thus the abstract properties of the categories constructed in this way justify
the intuition and terminology of “noncommutative geometry” which was one motivation for M. Kontsevich’s project of Noncommutative Motives and became the
central subject of Tabuada’s book.
This shift of the viewpoint required much work needed to establish how much
do we lose by passing from the classical picture to the new one, and what do we
gain in understanding of both old and new universes of Algebraic Geometry. Some
of these exciting results are surveyed in Tabuada’s monograph, and the reader who
wants to focus on a particular strand of research will be able to follow the relevant
original papers cited in the ample references list.
This stimulating book will be a precious source of information for all researchers
interested in algebraic geometry.
Yuri I. Manin
Introduction
The theory of motives began in the early sixties when Grothendieck envisioned
the existence of a “universal cohomology theory of algebraic varieties” acting as a
gateway between algebraic geometry and the assortment of the classical Weil cohomology theories (de Rham, Betti, étale, crystalline). After the release of Manin’s
foundational article [Man68] on the subject, Grothendieck’s ideas became popular
and a powerful driving force in mathematics.
The theory of noncommutative motives is more recent. It began in the eighties
when the Moscow school (Beilinson, Bondal, Kapranov, Manin, and others) started
the study of algebraic varieties via their derived dg categories of coherent sheaves.
It turns out that several invariants of algebraic varieties can be recovered from
their derived dg categories. The idea of replacing algebraic varieties by arbitrary
dg categories, which are morally speaking “noncommutative algebraic varieties”,
later led Kontsevich [Konb] to envision the existence of a “universal invariant of
noncommutative algebraic varieties”.
The purpose of this book is to give a rigorous overview of some of the main
advances in the theory of noncommutative motives. It is based on a graduate course
(18.917 - Noncommutative Motives) taught at MIT in the spring of 2014 and its
intended audience consists of graduate students and mathematicians interested in
noncommutative motives and their applications. We assume some familiarity with
algebraic geometry and with homological/homotopical algebra. The contents of the
book can be divided into three main parts:
• Part I: Differential graded categories – Chapter 1.
• Part II: Noncommutative pure motives – Chapters 2-7.
• Part III: Noncommutative mixed motives – Chapters 8-10.
A differential graded (=dg) category is a category enriched over complexes (morphism sets are complexes). An essential example to keep in mind is the derived
dg category of an algebraic variety. Several invariants such as algebraic K-theory,
cyclic homology (and all its variants), and topological Hochschild homology, can be
defined directly on dg categories. In order to study all these invariants simultaneously, we introduce the notion of an additive invariant and of a localizing invariant.
A functor, defined on the category of (small) dg categories and with values in
an additive category, is called an additive invariant if it inverts Morita equivalences
and sends semi-orthogonal decompositions in the sense of Bondal-Orlov to direct
sums. Chapter 2 is devoted to the study of this class of invariants and to the construction of the universal additive invariant. The theory of noncommutative pure
motives can be roughly summarized as the study of the target additive category of
this universal additive invariant. Making use of Kontsevich’s smooth proper dg categories, which are morally speaking the “noncommutative smooth proper algebraic
1
2
INTRODUCTION
varieties”, we introduce in Chapter 4 several additive categories of noncommutative
pure motives and relate them to their commutative counterparts. An important
example is the category of noncommutative numerical motives. Among other properties, we prove that this category is abelian semi-simple. This result, combined
with some noncommutative (standard) conjectures stated in Chapter 5, leads to a
(conditional) theory of noncommutative motivic Galois groups and to the extension of the classical theory of intermediate Jacobians to “noncommutative algebraic
varieties”; consult Chapters 6 and 7, respectively.
A functor, defined on the category of (small) dg categories and with values in
a triangulated category, is called a localizing invariant if it inverts Morita equivalences, preserves filtered homotopy colimits, and sends Drinfeld’s short exact sequences to distinguished triangles. The rigorous formalization of this notion requires the language of Grothendieck derivators, which can be found in Appendix A.
Chapter 8 is devoted to the study of this class of invariants and to the construction
of the universal localizing invariant. The theory of noncommutative mixed motives
can be roughly summarized as the study of the target triangulated category of
the universal localizing invariant. Making use once again of Kontsevich’s smooth
proper dg categories, we introduce in Chapter 9 several triangulated categories
of noncommutative mixed motives and relate them with Voevodsky’s triangulated
category of geometric mixed motives. Finally, in Chapter 10 we briefly describe the
(unconditional) theory of noncommutative motivic Hopf dg algebras, which is the
mixed analogue of the theory of noncommutative motivic Galois groups.
Although recent, the theory of noncommutative motives already led to the
(partial) solution of some open problems and conjectures in adjacent research areas.
Some of these will be discussed throughout the book.
We refrain from giving a lengthy introduction to the contents of each chapter.
The table of contents combined with the introduction of each chapter provides the
corresponding information.
Acknowledgements: My first words of gratitude are reserved to Yuri I.
Manin. I acknowledge and thank him for introducing me to the beautiful theory
of motives through his writings, in particular via his foundational article [Man68].
Notwithstanding his many solicitations, he has always been very generous with his
time and I profoundly thank him for honoring me with the preface of this book.
I would also like to thank the audience of the graduate course (18.917 - Noncommutative Motives) for pertinent questions and comments that helped shape the
way in which the material of this book is presented.
My acknowledgments also go to the Department of Mathematics at MIT for
the opportunity to teach this graduate course and to the National Science Foundation and the Fundação para a Ciência e a Tecnologia (Portuguese Foundation
for Science and Technology) for supporting my research, some of which is presented in this book, through the NSF CAREER Award #1350472 and the project
UID/MAT/00297/2013 (Centro de Matemática e Aplicações).
My final words of gratitude are reserved to my brother for meticulously guiding
me throughout the crossroads of life.
.