Table Of ContentMathematical
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
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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 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 categoricalcontext.
Followingtheidentificationofthecategoricalmotiviccohomologyasthehigher
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-Oguscohomology 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].
Thevariousrealizationsofanobjectinthemotiviccategoryallowonetorelate
and unite parallel phenomena in different cohomology theories. A central example
isBeilinson’smotivic polylogarithm,togetherwithits Hodge and´etalerealizations
(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 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 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-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-
latingourcategorytothemotivicLiealgebraofBlochandKriz,orthetriangulated
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
theluxuryofayearofundisturbedscholarshipinlivelymathematicalsurroundings,
forwhichIammostgrateful;IalsowouldliketothankNortheasternUniversityfor
theleaveofabsencewhichmadethatvisitpossible. Specialandheartfeltthanksare
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 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
