mixed motives - m. levine

Mixed Motives Marc Levine Mathematical Surveys and Monographs Volume 57 American Mathematical Society ΑΓΕΩΜΕ ΕΙΣΙΤΩ ΤΡΗΤΟΣ ΜΗ F O U N D E D 1 8 8 8 A M E R I C A N M A T H E M A T I C A L S O C I E T Y Editorial Board Georgia Benkart Howard A. Masur Tudor Stefan Ratiu, Chair Michael Renardy 1991 Mathematics Subject Classification. Primary 19E15, 14C25; Secondary 14C15, 14C17, 14C40, 19D45, 19E08, 19E20. Research supported in part by the National Science Foundation and the Deutsche Forschungsgemeinschaft. Abstract. The author constructs and describes a triangulated category of mixed motives over an arbitrary base scheme. The resulting cohomology theory satisfies the Bloch-Ogus axioms; if the base scheme is a smooth scheme of dimension at most one over a field, this cohomology theory agrees with Bloch’s higher Chow groups. Most of the classical constructions of cohomology can be made in the motivic setting, including Chern classes from higher K-theory, push-forward for proper maps, Riemann-Roch, duality, as well as an associated motivic homology, Borel-Moore homology and cohomology with compact supports. The motivic category admits a realization functor for each Bloch-Ogus cohomology theory which satisfies certain axioms; as examples the author constructs Betti, etale, and Hodge realizations over smooth base schemes. This book is a combination of foundational constructions in the theory of motives, together with results relating motivic cohomology with more explicit constructions, such as Bloch’s higher Chow groups. It is aimed at research mathematicians interested in algebraic cycles, motives and K-theory, starting at the graduate level. It presupposes a basic background in algebraic geometry and commutative algebra. Library of Congress Cataloging-in-Publication Data Levine, Marc, 1952– Mixed motives / Marc Levine. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 57) Includes bibliographical references and indexes. ISBN 0-8218-0785-4 (acid-free) 1. Motives (Mathematics) I. Title. II. Series: Mathematical surveys and monographs ; no. 57. QA564.L48 1998 516.3  5—dc21 98-4734 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams.org. c  1998 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. 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 URL: http://www.ams.org/ 10987654321 030201009998 iii To Ute, Anna, and Rebecca iv Preface This monograph is a study of triangulated categories of mixed motives over a base scheme S, whose construction is based on the rough ideas I originally outlined in a lecture at the J.A.M.I. conference on K-theory and number theory, held at the Johns Hopkins University in April of 1990. The essential principle is that one can form a categorical framework for motivic cohomology byfirstformingatensorcate- gory from the category of smooth quasi-projective schemes over S, with morphisms generated by algebraic cycles, pull-back maps and external products, imposing the relations of functoriality of cycle pull-back and compatibility of cycle products with the external product, then taking the homotopy category of complexes in this tensor category, and finally localizing to impose the axioms of a Bloch-Ogus cohomology theory, e.g., the homotopy axiom, the K¨unneth isomorphism, Mayer-Vietoris, and so on. Remarkably, this quite formal construction turns out to give the same coho- mology theory as that given by Bloch’s higher Chow groups [19], (at least if the base scheme is Spec of a field, or a smooth curve over a field). In particular, this puts the theory of the classical Chow ring of cycles modulo rational equivalence in a categorical context. Following the identification of the categorical motivic cohomology as the higher Chow groups, we go on to show how the familiar constructions of cohomology: Chern classes, projective push-forward, the Riemann-Roch theorem, Poincar´e du- ality, as well as homology, Borel-Moore homology and compactly supported coho- mology, have their counterparts in the motivic category. The category of Chow motives of smooth projective varieties, with morphisms being the rational equiva- lence classes of correspondences, embeds as a full subcategory of our construction. Our motivic category is specially constructed to give realization functors for Bloch-Ogus cohomology theories. As particular examples, we construct realization functors for classical singular cohomology, ´etale cohomology, and Hodge (Deligne) cohomology. We also have versions over a smooth base scheme, the Hodge realiza- tion using Saito’s category of algebraic mixed Hodge modules. We put the Betti, ´etale and Hodge relations together to give the “motivic” realization into the cate- gory of mixed realizations, as described by Deligne [32], Jannsen [71], and Huber [67]. The various realizations of an object in the motivic category allow one to relate and unite parallel phenomena in different cohomology theories. A central example is Beilinson’s motivic polylogarithm, together with its Hodge and ´etale realizations (see [9]and[13]). Beilinson’s original construction uses the weight-graded pieces of the rational K-theory of a certain cosimplicial scheme over P 1 minus {0, 1, ∞} as a replacement for the motivic object; essentially the same construction gives rise v vi PREFACE to the motivic polylogarithm as an object in our category of motives over P 1 minus {0, 1, ∞}, with the advantage that one acquires some integral information. There have been a number of other constructions of triangulated motivic cat- egories in the past few years, inspired by the conjectural framework for mixed mo- tives set out by Beilinson [10] and Deligne [32], [33]. In addition to the approach via mixed realizations mentioned above, constructions of triangulated categories of motives have been given by Hanamura [63]andVoevodsky[124]. Deligne has sug- gested that the category of Q-mixed Tate motives might be accessible via a direct construction of the “motivic Lie algebra”; the motivic Tate category would then be given as the category of representations of this Lie algebra. Along these lines, Bloch and Kriz [17] attempt to realize the category of mixed Tate motives as the category of co-representations of an explicit Lie co-algebra, built from Bloch’s cycle complex. Kriz and May [81] have given a construction of a triangulated category of mixed Tate motives (with Z-coefficients) from co-representations of the “May algebra” given by Bloch’s cycle complex. The Bloch-Kriz category has derived cat- egory which is equivalent to the Q-version of the triangulated category constructed by Kriz and May, if one assumes the Beilinson-Soul´e vanishing conjectures. We are able to compare our construction with that of Voevodsky, and show that, when the base is a perfect field admitting resolution of singularities, the two categories are equivalent. Although it seems that Hanamura’s construction should give an equivalent category, we have not been able to describe an equivalence. Re- lating our category to the motivic Lie algebra of Bloch and Kriz, or the triangulated category of Kriz and May, is another interesting open problem. Besides the categorical constructions mentioned above, there have been con- structions of motivic cohomology which rely on the axioms for motivic complexes set down by Lichtenbaum [90] and Beilinson [9], many of which rely on a motivic interpretation of the polylogarithm functions. This began with the Bloch-Wigner dilogarithm function, leading to a construction of weight two motivic cohomol- ogy via the Bloch-Suslin complex ([40]and[119]) and Lichtenbaum’s weight two motivic complex [89]. Pushing these ideas further has led to the Grassmann cy- cle complex of Beilinson, MacPherson, and Schechtman [15], as well as the mo- tivic complexes of Goncharov ([50], [51], [52]), and the categorical construction of Beilinson, Goncharov, Schechtman, and Varchenko [14]. Although we have the polylogarithm as an object in our motivic category, it is at present unclear how these constructions fit in with our category. While writing this book, the hospitality of the University of Essen allowed me the luxury of a year of undisturbed scholarship in lively mathematical surroundings, for which I am most grateful; I also would like to thank Northeastern University for the leave of absence which made that visit possible. Special and heartfelt thanks are due to H´el`ene Esnault and Eckart Viehweg for their support and encouragement. The comments of Spencer Bloch, Annette Huber, and Rick Jardine were most helpful and are greatly appreciated. I thank the reviewer for taking the time to go through the manuscript and for suggesting a number of improvements. Last, but not least, I wish to thank the A.M.S., especially Sergei Gelfand, Sarah Donnelly, and Deborah Smith, for their invaluable assistance in bringing this book to press. Boston Marc Levine November, 1997 Contents Preface v Part I. Motives 1 Introduction: Part I 3 Chapter I. The Motivic Category 7 1. The motivic DG category 9 2. The triangulated motivic category 16 3. Structure of the motivic categories 36 Chapter II. Motivic Cohomology and Higher Chow Groups 53 1. Hypercohomology in the motivic category 53 2. Higher Chow groups 65 3. The motivic cycle map 77 Chapter III. K-Theory and Motives 107 1. Chern classes 107 2. Push-forward 130 3. Riemann-Roch 161 Chapter IV. Homology, Cohomology, and Duality 191 1. Duality 191 2. Classical constructions 209 3. Motives over a perfect field 237 Chapter V. Realization of the Motivic Category 255 1. Realization for geometric cohomology 255 2. Concrete realizations 267 Chapter VI. Motivic Constructions and Comparisons 293 1. Motivic constructions 293 2. Comparison with the category DM gm (k) 310 Appendix A. Equi-dimensional Cycles 331 1. Cycles over a normal scheme 331 2. Cycles over a reduced scheme 347 Appendix B. K-Theory 357 1. K-theory of rings and schemes 357 2. K-theory and homology 360 vii viii CONTENTS Part II. Categorical Algebra 371 Introduction: Part II 373 Chapter I. Symmetric Monoidal Structures 375 1. Foundational material 375 2. Constructions and computations 383 Chapter II. DG Categories and Triangulated Categories 401 1. Differential graded categories 401 2. Complexes and triangulated categories 414 3. Constructions 435 Chapter III. Simplicial and Cosimplicial Constructions 449 1. Complexes arising from simplicial and cosimplicial objects 449 2. Categorical cochain operations 454 3. Homotopy limits 466 Chapter IV. Canonical Models for Cohomology 481 1. Sheaves, sites, and topoi 481 2. Canonical resolutions 486 Bibliography 501 Subject Index 507 Index of Notation 513 Part I Motives [...]... smooth projective S-scheme with a complement a normal crossing scheme We then examine extensions of the motivic theory to non-smooth S-schemes We give a construction of the Borel-Moore motive and the motive with compact support for certain non-smooth S-schemes; as an application we prove a RiemannRoch theorem for singular varieties We give a construction of the (cohomological) motive of k-scheme of finite... just for smooth varieties, but also for diagrams of smooth varieties We prove the Riemann-Roch theorem without 6 INTRODUCTION: PART I denominators, and the usual Riemann-Roch theorem As an application, we show that the Chern character gives an isomorphism of rational motivic cohomology with weight-graded K-theory, for motives over a field or a smooth curve over a field In Chapter IV we examine duality in... satisfying the associativity identity of a pseudo-functor Proof As is well known, the canonical isomorphism θp,q (X) : (p ◦ q)∗ (X) → q ∗ (p∗ (X)); X ∈ SchS , makes the operation of pull-back into a pseudo-functor The same identity thus holds for pull-back in the categories L(−) This then implies that sending p to the tensor functor A2 (p∗ ) (2.3.2.5) defines a pseudo-functor to tensor categories Using the functoriality... the universal cohomology theory of algebraic varieties is the category of mixed motives This category has yet to be constructed, although many of its desired properties have been described (see [10] and [1], especially [70]) Here is a partial list of the expected properties: 1 For each scheme S, one has the category of mixed motives over S, MMS ; MMS is an abelian tensor category with a duality involution... DM(S)pr of DM(S) generated by smooth projective S-schemes in SmS Combined with the operation of cup-product by cycle classes, this gives the action of correspondences as homomorphisms in the category DM(S), and leads to a fully faithful embedding of the category of graded Chow motives (over a field k) into DM(Spec k) We define the homological motive, the Borel-Moore motive and the compactly supported motive... isomorphism to define the product, the groups u p Hµ (X, Z(q)) := Extp S (Z(0), M (X) ⊗ Z(q)) MM form a bi-graded ring which satisfies the axioms of a Bloch-Ogus cohomology theory: Mayer-Vietoris for Zariski open covers, homotopy property, projective bundle formula, etc 6 There are Chern classes from algebraic K-theory p cq,p : K2q−p (X) → Hµ (X, Z(q)) which induce an isomorphism K2q−p (X)(q) ∼ Hµ (X, Z(q))... case the base is a field, or is a smooth curve over a field, the Chern character defines an isomorphism of rational motivic cohomology with weight-graded K-theory, as required by (6) For a Bloch-Ogus twisted duality theory Γ, defined via cohomology of a complex of A-valued sheaves for a Grothendieck topology T on SmS , satisfying certain natural axioms, the motivic triangulated category DM(S) admits a realization... (−)(q) the weight q eigenspace of the Adams operations p 7 The cohomology theory Hµ (X, Z(q)) is universal: Each Bloch-Ogus coho∗ mology theory X → H (X, Γ(∗)) gives rise to a natural transformation ∗ Hµ (−, Z(∗)) → H ∗ (−, Γ(∗)) 8 MMS ⊗ Q is a Tannakian category, with the Q-Betti or Ql - tale realization e giving a fiber functor 3 4 INTRODUCTION: PART I 9 There is a natural weight filtration on the objects... theory from that of Bloch-Ogus [20] or Gillet [46], but it seems that this type of cohomology theory is general enough for many applications We construct the Betti, ´tale and e Hodge realizations of DM(V) in subsequent sections; we also give the realization to Saito’s category of mixed Hodge modules [110] (over a smooth base) and to a version of Jannsen’s category [71] of mixed absolute Hodge complexes... Continue inverting maps until the various axioms of a Bloch-Ogus cohomology theory are satisfied (vi) This forms a triangulated tensor category; take the pseudo-abelian hull to give the triangulated tensor category DM(S) There are several problems with this naive approach The first is that the relation (iii) is only given for cycles Z for which the pull-back p∗ (Z) is defined Classically, this type of problem . Mixed Motives Marc Levine Mathematical Surveys and Monographs Volume 57 American Mathematical Society ΑΓΕΩΜΕ ΕΙΣΙΤΩ ΤΡΗΤΟΣ ΜΗ F O U N D E D 1 8 8 8 A M E R I C A N M A T H E M A T I C A L S O C I E T Y Editorial. S) M :(Sm/S) op →MM S , where Sm/S is the category of smooth S-schemes; M( X)isthemotive of X. 3. There are external products M( X) ⊗ M( Y ) → M (X × S Y ) which are isomorphisms (the K¨unneth isomorphism). 4 algebra. Library of Congress Cataloging-in-Publication Data Levine, Marc, 1952– Mixed motives / Marc Levine. p. cm. — (Mathematical surveys and monographs, ISSN 007 6-5 376 ; v. 57) Includes bibliographical