University LECTURE Series Volume 63 Noncommutative Motives Gonçalo Tabuada American Mathematical Society Noncommutative Motives http://dx.doi.org/10.1090/ulect/063 University LECTURE Series Volume 63 Noncommutative Motives Gonçalo Tabuada American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jordan S. Ellenberg Robert Guralnik William P. Minicozzi II (Chair) Tatiana Toro 2010 Mathematics Subject Classification. Primary 14A22, 14C15, 18D20; Secondary 18E30, 18G55, 19D55. For additional information and updates on this book, visit www.ams.org/bookpages/ulect-63 Library of Congress Cataloging-in-Publication Data Tabuada, Gon¸calo, 1979– Noncommutative motives / Gon¸calo Tabuada. pages cm. – (University lecture series ; volume 63) Includes bibliographical references and index. ISBN 978-1-4704-2397-1 (alk. paper) 1. Motives (Mathematics) 2. Noncommutative algebras. 3. Algebraic varieties. I. Title. QA564.T33 2015 516.35–dc23 2015018204 Copying and reprinting. 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Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2015 by Gon¸calo Tabuada. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 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 Cnc 57 5.2. Standard conjecture of type Dnc 58 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)ofsmoothk-varieties to an abelian tensor category Mot(k) of “(pure) motives” (or Mot(k)R,whereR 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. 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,orevenanR-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- tersection 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. 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,orKaroubian completion of the category. In this new category, the projective line P1 becomes the direct sum 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,notjust 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 project resists any simple-minded notion of “unification”, and with time, the theory of motives was more and more resembling a Borgesian garden ix xPREFACE 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. The book by Gon¸calo 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 only with 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 no longer sufficient: 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 non- necessarily commutative rings A → B are replaced with (A, B)-bimodules, and that the difference between commutative and noncommutative rings in this frame- work essentially vanishes 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 triangu- lated 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 tri- angulated 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 mo- tivation for M.