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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.

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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

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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, 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. 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 conference “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 framework 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 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 mo- tivation 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 to establish how much we lose by passing from the classical picture to the new one, and what we gain in understanding both the 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

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⊗-nilpotent motives, 35, 47, 53, 59 filtered homotopy colimit, 19, 72, 77, 80, (finite) I-cell, 6, 73, 102 102, 103 (super-)Tannakian formalism, 38, 64, 65 filtered homotopy colimits, 73 étale K-theory, 24, 72, 79 finite dimensional algebra of finite global dimension, 16, 32, 55 absolute Galois group, 39, 67, 98 full exceptional collection, 30, 44, 45, 56 additive A1-invariant, 79, 85 additive invariant, 21, 77, 85 Gaussian polynomial, 32 algebraic K-theory, 22, 77, 82, 89, 95 Amitsur’s conjecture, 31 higher Chern characters, 85 Artin motives, 37, 52 Hochschild homology, 25, 33, 34, 43, 72, 88 Ayoub’s weak Tannakian formalism, 97 Hochschild homology with coefficients, 33, 48 base-change functor, 53, 54, 66, 98 homological motives, 36, 48, 53 base-change short exact sequence, 39, 66, homotopically finitely presented dg 98 category, 18, 78, 82 binomial ring, 54–56 homotopy K-theory, 24, 72, 79, 82, 89, 95 blow-up, 29 integer coefficients, 50, 53 Bondal-Kapranov’s pretriangulated intermediate (algebraic) Jacobians, 69, 70 envelope, 14, 18, 22 intersection bilinear pairings, 69 Brauer group, 29, 30, 56, 89 Kahn-Levine’s twisted form of K-theory, 91 Calkin algebra, 75 Karoubi-Villamayor K-theory, 23, 77, 79, categorical trace, 17, 33, 36, 48 82, 89, 95 central simple algebra, 29–31, 56 Kimura-finiteness, 37, 59, 60, 94 change of coefficients, 51, 53 Kontsevich’s smooth proper dg category, Chow motives, 35, 41, 44, 46, 53, 69 16, 19, 33, 49, 50, 57–59, 65, 66, 74, 87 complex of exact cubes, 98 Kummer motives, 92 convolution product, 84 Künneth projectors, 37, 57, 63 corestriction, 56 cyclic homology, 25, 72 Lefschetz motive, 35, 44 Lefschetz’s formula, 32–34 de Rham cohomology, 36, 37, 47, 57, 69 left Bousfield localization, 14, 24, 73, 78, dg cluster category, 81 80, 101, 102 dg orbit category, 80 left properness, 10, 15, 102 differential operators in positive Levine’s mixed motives, 93 characteristic, 31 localizing A1-invariant, 79, 85 Drinfeld’s dg quotient, 11–13, 71, 73 localizing invariant, 72, 85 dualizable object, 17, 18, 33, 34, 74, 79 Dubrovin’s conjecture, 46 Mayer-Vietories for open covers, 76 Merkurjev-Panin’s motives, 46, 47 Euler characteristic, 17, 33, 38, 64 Milnor K-theory, 92 mixed Artin motives, 93 Fano threefolds, 46 mixed complex, 24, 25, 34, 72

113 114 INDEX

mixed Tate motives, 92 standard conjecture of type Cnc, 57, 60, 63, mod-lν algebraic K-theory, 23, 79, 83 65, 66 monoidal structure, 11, 16, 28, 74, 78, 80, standard conjecture of type D,37,58,60 81 standard conjecture of type Dnc,58,60, Morel-Voevodsky’s stable A1-homotopy 64–66 category, 90 Morita equivalence, 15, 21, 72, 74, 80 Tate motive, 35, 42, 92 motivic decompositions, 44 Tate motives, 36, 65 motivic Galois (super-)group, 38, 65–67 topological cyclic homology, 26, 73, 78 motivic measure, 42 topological Hochschild homology, 26, 73 motivic zeta function, 44 toric variety, 47 trace maps, 85, 86 negative Chern characters, 86 twisted flag variety, 31 negative cyclic homology, 73, 78 twisted Grassmannian variety, 31 nilpotent ideal, 32, 35 twisted projective homogeneous variety, 32 Nisnevich descent, 75 Nisnevich topology, 75 unconditional motivic Galois group, 66 noncommutative Chow motives, 41–43, 47, unconditional noncommutative motivic 69, 70, 89, 90 Galois group, 66 A1 noncommutative nilpotence conjecture, 59 universal additive -invariant, 81, 95 nonconnective algebraic K-theory, 23, 72, universal additive invariant, 27, 78, 79, 95 A1 82, 89, 95 universal localizing -invariant, 80, 81, 95 number field, 94 universal localizing invariant, 73, 75, 76, 95 numerical motives, 36, 48, 50, 53, 65 Voevodsky’s geometric mixed motives, orbit category, 41, 42, 92 91–94 Voevodsky’s nilpotence conjecture, 37 pairings, 83, 84 periodic Chern characters, 86 Waldhausen’s S•-construction, 79 periodic complex, 72 weight spectral sequence, 90 Weil cohomology theory, 36, 45 periodic cyclic homology, 25, 34, 47, 57, 64, 65, 73, 78, 80 phantom, 32 Picard group, 89 pretriangulated dg category, 13, 14, 30, 77 projective bundle, 29 purely inseparable field extension, 30 quadric fibration, 29, 59, 70 quasi-compact quasi-separated scheme, 12, 22, 29, 31, 71, 75, 76, 82, 84 rational coefficients, 88 rational Witt ring, 84 right properness, 10, 15

Schur-ﬁniteness, 37, 59, 60, 94 semi-orthogonal decomposition, 21, 29 semi-simplicity, 36, 50, 52, 66, 70 separable algebra, 47, 52, 53, 93 Serre functor, 49 Severi-Brauer variety, 29–31 sheaf of Azumaya algebras, 31 short exact sequence of dg categories, 71–73 sign conjecture, 37, 57, 60 smooth proper dg algebra, 18, 33, 49, 54, 82 split short exact sequence of dg categories, 77, 78 standard conjecture of Lefschetz type, 69 standard conjecture of type C,37,57,60 Selected Published Titles in This Series

63 Gon¸calo Tabuada, Noncommutative Motives, 2015 62 H. Iwaniec, Lectures on the Riemann Zeta Function, 2014 61 Jacob P. Murre, Jan Nagel, and Chris A. M. Peters, Lectures on the Theory of Pure Motives, 2013 60 William H. Meeks III and Joaqu´ın Pérez, A Survey on Classical Minimal Surface Theory, 2012 59 Sylvie Paycha, Regularised Integrals, Sums and Traces, 2012 58 Peter D. Lax and Lawrence Zalcman, Complex Proofs of Real Theorems, 2012 57 Frank Sottile, Real Solutions to Equations from Geometry, 2011 56 A. Ya. Helemskii, Quantum Functional Analysis, 2010 55 Oded Goldreich, A Primer on Pseudorandom Generators, 2010 54 John M. Mackay and Jeremy T. Tyson, Conformal Dimension, 2010 53 John W. Morgan and Frederick Tsz-Ho Fong, Ricci Flow and Geometrization of 3-Manifolds, 2010 52 Marian Aprodu and Jan Nagel, Koszul Cohomology and Algebraic Geometry, 2010 51 J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and Bálint Virág, Zeros of Gaussian Analytic Functions and Determinantal Point Processes, 2009 50 John T. Baldwin, Categoricity, 2009 49 József Beck, Inevitable Randomness in Discrete Mathematics, 2009 48 Achill Schürmann, Computational Geometry of Positive Definite Quadratic Forms, 2008 47 Ernst Kunz, David A. Cox, and Alicia Dickenstein, Residues and Duality for Projective Algebraic Varieties, 2008 46 Lorenzo Sadun, Topology of Tiling Spaces, 2008 45 Matthew Baker, Brian Conrad, Samit Dasgupta, Kiran S. Kedlaya, and Jeremy Teitelbaum, p-adic Geometry, 2008 44 Vladimir Kanovei, Borel Equivalence Relations, 2008 43 Giuseppe Zampieri, Complex Analysis and CR Geometry, 2008 42 Holger Brenner, Jürgen Herzog, and Orlando Villamayor, Three Lectures on Commutative Algebra, 2008 41 James Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics, 2008 40 Vladimir Pestov, Dynamics of Infinite-dimensional Groups, 2006 39 Oscar Zariski, The Moduli Problem for Plane Branches, 2006 38 Lars V. Ahlfors, Lectures on Quasiconformal Mappings, Second Edition, 2006 37 Alexander Polishchuk and Leonid Positselski, Quadratic Algebras, 2005 36 Matilde Marcolli, Arithmetic Noncommutative Geometry, 2005 35 Luca Capogna, Carlos E. Kenig, and Loredana Lanzani, Harmonic Measure, 2005 34 E. B. Dynkin, Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations, 2004 33 Kristian Seip, Interpolation and Sampling in Spaces of Analytic Functions, 2004 32 Paul B. Larson, The Stationary Tower, 2004 31 John Roe, Lectures on Coarse Geometry, 2003 30 Anatole Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, 2003 29 Thomas H. Wolff, Lectures on Harmonic Analysis, 2003 28 Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre, Cohomological Invariants in Galois Cohomology, 2003

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/ulectseries/. The theory of motives began in the early 1960s when Grothendieck envisioned the existence of a “universal cohomology theory of algebraic varieties”. The theory of noncommutative motives is more recent. It began in the 1980s when the Moscow school (Beilinson, Bondal, Kapranov, Manin, and others) began the study of algebraic varieties via their derived categories of coherent sheaves, and continued in the 2000s when Kontsevich conjectured the existence of a “universal invariant of noncommutative algebraic varieties”. This book, prefaced by Yuri I. Manin, gives a rigorous overview of some of the main advances in the theory of noncommutative motives. It is divided into three main parts. The ﬁrst part, which is of independent interest, is devoted to the study of DG categories from a homotopical viewpoint. The second part, written with an emphasis on examples and applications, covers the theory of noncommutative pure motives, noncommutative standard conjectures, noncommutative motivic Galois groups, and also the relations between these notions and their commutative counterparts. The last part is devoted to the theory of noncommutative mixed motives. The rigorous formalization of this latter theory requires the language of Grothendieck derivators, which, for the reader’s convenience, is revised in a brief appendix.

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