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Mathematical Surveys and Monographs Volume 57 Mixed Motives Marc Levine ERMIACAN MΑΓΕΩΜΕAΤTΡHΗEΤΟMΣ AΜTΗICΕΙΣΙΤΩALSOYCTIE American Mathematical Society FOUNDED 1888 Editorial Board Georgia Benkart TudorStefan Ratiu, Chair Howard A.Masur Michael Renardy 1991 Mathematics Subject Classification. Primary 19E15, 14C25; Secondary 14C15, 14C17, 14C40, 19D45, 19E08, 19E20. Research supported in part bythe National Science Foundation and theDeutsche Forschungsgemeinschaft. Abstract.Theauthorconstructsanddescribesatriangulatedcategoryofmixedmotivesoveran arbitrarybase 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 authorconstructs Betti,etale,andHodgerealizationsoversmoothbaseschemes. This book is a combination of foundational constructions in the theory of motives, together withresultsrelatingmotiviccohomologywithmoreexplicitconstructions,suchasBloch’shigher Chowgroups. Itisaimedatresearchmathematicians interested inalgebraiccycles, motives and K-theory,startingatthegraduatelevel. Itpresupposesabasicbackgroundinalgebraicgeometry andcommutativealgebra. Library of Congress Cataloging-in-PublicationData Levine,Marc,1952– Mixedmotives/MarcLevine. p.cm. —(Mathematical surveysandmonographs,ISSN0076-5376;v.57) Includes bibliographicalreferencesandindexes. ISBN0-8218-0785-4(acid-free) 1. Motives (Mathematics) I. Title. II. Series: Mathematical surveys and monographs ; no.57. QA564.L48 1998 516.3(cid:1)5—dc21 98-4734 CIP Copying and reprinting. Individual readersof thispublication, and nonprofitlibrariesacting 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,providedthecustomaryacknowledgment ofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication (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 [email protected]. (cid:1)c 1998bytheAmericanMathematical Society. Allrightsreserved. TheAmericanMathematical Societyretainsallrights exceptthosegrantedtotheUnitedStates Government. PrintedintheUnitedStates ofAmerica. (cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageatURL: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 baseschemeS,whoseconstructionis basedonthe roughideasI originallyoutlined inalectureatthe J.A.M.I.conferenceonK-theoryandnumbertheory,heldatthe Johns Hopkins University in April of 1990. The essential principle is that one can formacategoricalframeworkformotivic cohomology byfirstformingatensorcate- goryfromthecategoryofsmoothquasi-projectiveschemesoverS,withmorphisms generatedby algebraiccycles, pull-back maps and external products, imposing the relationsoffunctorialityofcyclepull-backandcompatibilityofcycleproductswith theexternalproduct,thentakingthehomotopycategoryofcomplexesinthistensor category, and finally localizing to impose the axioms of a Bloch-Ogus cohomology theory, e.g., the homotopy axiom, the Ku¨nneth isomorphism, Mayer-Vietoris, and so on. Remarkably, this quite formal construction turns out to give the same cohomology 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 categoricalcontext. Followingtheidentificationofthecategoricalmotiviccohomologyasthehigher Chow groups, we go on to show how the familiar constructions of cohomology: Chern classes, projective push-forward, the Riemann-Roch theorem, Poincaré duality, as well as homology, Borel-Moore homology and compactly supported cohomology, have their counterparts in the motivic category. The category of Chow motives of smooth projective varieties, with morphisms being the rational equivalence classes of correspondences,embeds as a full subcategory of our construction. Our motivic category is specially constructed to give realization functors for Bloch-Oguscohomology theories. As particular examples, we construct realization functors for classical singular cohomology,étale cohomology, and Hodge (Deligne) cohomology. We also have versions over a smooth base scheme, the Hodge realization using Saito’s category of algebraic mixed Hodge modules. We put the Betti, étale and Hodge relations together to give the “motivic” realization into the category of mixed realizations, as described by Deligne [32], Jannsen [71], and Huber [67]. Thevariousrealizationsofanobjectinthemotiviccategoryallowonetorelate and unite parallel phenomena in different cohomology theories. A central example isBeilinson’smotivic polylogarithm,togetherwithits Hodge andétalerealizations (see [9] and [13]). Beilinson’s original construction uses the weight-graded pieces of the rational K-theory of a certain cosimplicial scheme over P1 minus {0,1,∞} asareplacementforthe motivicobject;essentiallythe sameconstructiongivesrise v vi PREFACE tothe motivicpolylogarithmasanobjectinourcategoryofmotivesoverP1 minus {0,1,∞},with the advantage that one acquires some integral information. There have been a number of other constructions of triangulated motivic categories in the past few years, inspired by the conjectural framework for mixed motives set out by Beilinson [10] and Deligne [32], [33]. In addition to the approach via mixedrealizationsmentionedabove,constructionsof triangulatedcategoriesof motiveshavebeengivenby Hanamura[63]andVoevodsky[124]. Deligne hassug- 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 categoryofco-representationsofanexplicitLieco-algebra,builtfromBloch’scycle 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”givenbyBloch’scycle complex. The Bloch-Krizcategoryhasderivedcat- egorywhichis equivalentto the Q-versionofthe triangulatedcategoryconstructed by Kriz and May, if one assumes the Beilinson-Soulé 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- latingourcategorytothemotivicLiealgebraofBlochandKriz,orthetriangulated category of Kriz and May, is another interesting open problem. Besides the categorical constructions mentioned above, there have been constructions 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 cohomology 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 cycle complex of Beilinson, MacPherson, and Schechtman [15], as well as the motivic 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 theluxuryofayearofundisturbedscholarshipinlivelymathematicalsurroundings, forwhichIammostgrateful;IalsowouldliketothankNortheasternUniversityfor theleaveofabsencewhichmadethatvisitpossible. Specialandheartfeltthanksare due to Hélène 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 categoryDM (k) 310 gm 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. Categoricalcochain 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