Selected Titles in This Series

57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum

groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in

analysis, 1997 53 Andreas Kriegl and Peter W . Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in

domains with point singularities, 1997 51 Jan Maly and Wil l iam P. Ziemer, Fine regularity of solutions of elliptic partial

differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential

equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert , Integer-valued polynomials, 1997 47 A. D . Elmendorf, I. Kriz, M. A. Mandell , and J. P. May (with an appendix by

M. Cole) , Rings, modules, and algebras in stable homotopy theory, 1997 46 S tephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and A d a m O. Hausknecht, Cogroups and co-rings in

categories of associative rings, 1996 44 J. Amoros , M. Burger, K. Corlette , D . Kotschick, and D . Toledo, Fundamental

groups of compact Kahler manifolds, 1996 43 James E. Humphreys , Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nat ion , Free lattices, 1995 41 Hal L. Smith , Monotone dynamical systems: an introduction to the theory of

competitive and cooperative systems, 1995 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 3, 1998 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 G u y David and Stephen Semmes , Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean

fields, 1990 32 Howard Jacobowitz , An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and

harmonic analysis on semisimple Lie groups, 1989 30 Thomas W . Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals , Percy Deift , and Carlos Tomei, Direct and inverse scattering on the

line, 1988 27 Na than J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici , Operator theory and arithmetic in H°°, 1988

(Continued in the back of this publication)

http://dx.doi.org/10.1090/surv/057

Mathematical Surveys

and Monographs

Volume 57

glEMgg.

Mixed Motives

Marc Levine

American Mathematical Society

Editorial Board Georgia Benkart Tudor Stefan Ra t iu , Chair Howard A. Masur Michael Renardy

1991 Mathematics Subject Classification. P r i m a r y 19E15, 14C25; Secondary 14C15, 14C17, 14C40, 19D45, 19E08, 19E20.

Research suppor ted in pa r t by the Nat ional Science Foundat ion 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 X-theory, starting at the graduate level. It presupposes a basic background in algebraic geometry and commutative algebra.

Library of Congress Cataloging-in-Publicat ion D a t a 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

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10 9 8 7 6 5 4 3 2 1 03 02 01 00 99 98

To Ute, Anna, and Rebecca

Contents

Preface ix

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 DMgm(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

viii C O N T E N T S

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

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 by first forming a tensor cate­gory from the category of smooth quasi-projective schemes over 5, 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 Kunneth isomorphism, Mayer-Viet oris, 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 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, Poincare 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-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 realiza­tion 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 cate­gory 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 poly logarithm, together with its Hodge and etale realizations (see [9] and [13]). Beilinson's original construction uses the weight-graded pieces of the rational if-theory of a certain cosimplicial scheme over P1 minus {0,1, oo} as a replacement for the motivic object; essentially the same construction gives rise

ix

X PREFACE

to the motivic polylogarithm as an object in our category of motives over P1 minus {0, l ,oo}, 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 mixed realizations mentioned above, constructions of triangulated categories of motives have been given by Hanamura [63] and Voevodsky [124]. Deligne has sug­gested 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 cat­egory which is equivalent to the Q-version of the triangulated category constructed by Kriz and May, if one assumes the Beilinson-Soule 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. Re­lating 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 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 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 Helene 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 November, 1997

Marc Levine

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Subject Index

A absolute Hodge complexes, 273-275

enlarged diagrams, 274-275 Adams degree, 258 additive category, 381

free, 384 adjoining a base-point, 361 adjoining morphisms

to a category, 384 to a DG tensor category, 422-424 to a tensor category, 393

Alexander-Whitney map, 463, 478

B bi-products, 380 Bloch's formula, 93 blow-up distinguished triangle, 237-239

C canonical cochains, 258 canonical filtration, 277 canonical topology, 482 canonical truncation, 461 categorical cochain operations, 464-466 category of pairs, 385 Cech resolution, 53, 56 Chern character, 169-171

for diagrams, 170 for higher if-theory, 164 isomorphism, 179, 188

Chern classes compatibility with localization and rela-

tivization, 126 for K0i 163 for higher K-theory, 119, 122, 164

mod n, 124, 126 of a diagram, 122-126 total, 165 with support, 123

for line bundles, 112 properties, 112

for relative higher if-theory, 124 with support, 124

for vector bundles, 116 naturality, 117 properties, 116 total, 116, 164

universal, 121 Chow motives, 214-215

definition, 214 embedding into PA4, 215

Chow's lemma, 237 classifying scheme, 357-358

bundles, 358 classifying space, 357 closed simplicial model category, 475 cocontinuous, 255 codegeneracy map, 449 coface map, 449 cohomology over a category, 466, 467 compactifiable embedding, 225 compactification, 277 complexes over a DG category, 409, 411-414

cone sequence, 416 homotopy category, 411

distinguished triangles, 416 triangulated structure, 416-420

tensor structure, 414 cone

for complexes over a DG category, 411 for complexes over an additive category,

410 for Pre-Tr, 411

cone sequence for complexes over a DG category, 416 for complexes over an additive category,

410 connected

by a subset, 335 in codimension one, 335

connectivity, 335 correspondences, 210, 314 cosimplicial scheme

motive of a, 26 very smooth, 27

coskeleton, 482 cup product, 453 cycle class map, 47-51

compatibility with Gysin morphism, 138 for K-theory, 183-185 for units, 294 for varieties, 48 motivic, 76-78, 82

507

508 SUBJECT INDEX

injectivity, 84 isomorphism, 88 surjectivity, 78, 82

naive, 71, 76 properties, 48-51 relative, 218

push-forward for, 218 with support, 48

for varieties, 48 cycle complex

Bloch's, 65, 68 comparison isomorphism, 68 motivic, 68, 69

for varieties, 77 cycle map, 47-51

for varieties, 48 with support, 17

for varieties, 48 cycles

and cycle classes, 47 basic definition, 10 effective, 332 equi-dimensional

properties of, 352-356 pull-back for, 346

for £(V), 10 functoriality, 10 intersection multiplicity, 333

over a normal base, 340 over a reduced base, 349

of relative dimension d, 332 on simplicial schemes, 107, 110

products, 109 relative, 181, 331

cycles functor, 13, 38 construction, 38-40

D £>-module

holonomic, 282 de Rham functor, 282 decomposition into type, 445 deformation diagram, 131 differential graded category, 381

homotopy equivalence, 420 dimension, 331

over a scheme, 331 discriminant, 239 distinguished octahedra, 429 Dold-Kan equivalence, 362 dual

canonical, 194 of a morphism, 193 of an object, 192

duality in tensor categories, 191 in triangulated tensor categories, 198

duality criterion for a tensor category, 195-197

for a triangulated tensor category, 204 duality involution

explicit formulae, 210-214 for a graded tensor category, 195 for a tensor category, 194 for a triangulated tensor category, 201,

204 for smooth projective schemes, 205-206 for the triangulated motivic category, 206,

207 for the triangulated Tate category, 235

decalage, 278

E effective motives

category of, 311 tensor product for, 312

Eilenberg-MacLane map, 477 equi-dimensional

cycle, 332 scheme, 331

etale site, 482 exact category, 358 excision isomorphism, 18 extended total complex, 455 external product, 391

categorical, 391 universal, 392

F fiber functor, 483

associated pro-object, 485 stalk, 484

fibrant functor, 475 simplicial set, 475

finite category, 468 flat

inverse system, 271 presheaf, 256 sheaf, 256

flatness, 497

G general linear group, 358 generic projection, 95 geometric cohomology theory, 255, 257 geometric motives

category of, 313 geometric point, 269, 331 gluing cycles, 347 Godement resolution, 486, 490-499

and cohomology with support, 490, 492 and flatness, 499 and sheaf cohomology, 490 associated complex, 490 products, 493, 496

good compactifications, 224 duality for, 224

SUBJECT INDEX 509

graded category, 381 graded homomorphism, 383 graded symmetric monoidal category, 436

punctual, 436 Grothendieck group, 359 Grothendieck pre-topology, 481 Grothendieck site, 481

presheaf on a, 482 sheaf on a, 482

Grothendieck topology, 481 covering families, 481

group homology, 357 Gysin distinguished triangle, 132 Gysin isomorphism, 18 Gysin morphism, 20, 130, 141

compatibility with cycle classes, 138 compatibility with products, 142 for a closed embedding, 131-132 for a split embedding, 130-131 functoriality, 132 properties, 132

Gysin sequence, 132

H higher Chow groups

and motivic cohomology, 103, 105 Bloch's, 65, 66

properties, 66 comparison isomorphism, 71 motivic, 75-77

and K-theory, 179 and hypercohomology, 77 for varieties, 77

naive, 70 for varieties, 72

homotopy category of a DG category, 409 of the category of complexes, 411

homotopy commutative product, 463 homotopy commutativity, 322, 441, 445, 455 homotopy equivalence of DG categories, 420 homotopy fiber, 127 homotopy limit

additive, 467, 468, 471 and cohomology, 473 and hypercohomology, 474 distinguished triangle, 472 functoriality, 472 non-degenerate, 471

for simplicial sets, 474-475 homotopy one point category, 13, 435, 440-

441 universal mapping property, 441

homotopy property, 17 for homological motives, 216 for motives of diagrams, 35 for motives of schemes, 19

Hurewicz map, 362-363 compatibility with products, 364-369

for diagrams, 363-364 hypercohomology, 62

for motives, 63, 64 hypercover, 482

of a sheaf, 483 of an object, 483

hyperresolutions cubical, 237, 240

category of, 240 strict, 240 weak, 247

for motives, 59 maps of, 59 the category of, 59, 61

I index of inseparability, 338 intersection multiplicity

on a regular scheme, 333 over a normal base, 340 over a reduced base, 349

inverse systems category of, 269-270

internal Horn, 270 tensor structure, 270

continuous hypercohomology for, 270 of sheaves, 269 of Tate sheaves, 270 strongly acyclic, 271 strongly normalized, 271

K K- theory

and homology of GL, 362 for diagrams, 360 for schemes, 358

K-group relative, 123

with support, 124 with support, 123

Kiinneth isomorphism, 18 for compactly supported motives, 217 for diagrams, 36 for homological motives, 216

L L-functions of motives, 292 lambda ring, 161

Adams operations, 180 for higher if-theory, 181

augmented, 180 gamma filtration, 180 special, 162-163 structure for KQ, 163 structure for higher K-theory, 179-180

Leibniz rule, 295 localization

connecting homomorphism, 299—302 distinguished triangle, 22

510 SUBJECT INDEX

for homological motives, 216 for motives with support, 22 for relative motives, 34

of a triangulated category, 425-426 of a triangulated tensor category, 426 sequence, 126, 129

compatibility with Chern classes, 128-130

M MAH complex, 285 Mayer-Vietoris

for homological motives, 216 for motives, 21 for motives with support, 22

Milnor K-groups, 293, 298-303 and motivic cohomology, 298 tame symbol, 302

Milnor K-sheaf, 303 mixed absolute Hodge complex, 285 mixed Hodge modules, 282 monoidal category, 375 motive

Cech resolution, 54, 58 Borel-Moore, 153

cap products, 217 for singular schemes, 225, 227-231 functoriality, 153 properties, 217 with support, 153, 226

compactly supported, 215 cup products, 217 for singular schemes, 227-231 properties, 217

fundamental properties, 19 homological, 215

cap products, 217 projection formula, 217 properties, 216-217 relative, 216 Thorn isomorphism, 216

Lefschetz, 23 of a /c-scheme, 241

Borel-Moore, 251-253 cohomological, 246-249 compactly supported, 251-253 comparisons, 250 homological, 249-250 products, 248

of a cosimplicial scheme, 27-28 of a cubical hyperresolution, 241

blow-up diagram for, 241 of a diagram, 34-35

distinguished triangle, 35-36 products, 36 properties, 156

of a non-degenerate simplicial scheme, 30 of a simplicial scheme, 28-31 of a variety, 21

of an n-cube, 31-32 open cover, 54, 58

push-forward, 56 relative, 31, 33-34, 123

duality for, 219-224 with support, 34, 124

with support, 21 of a diagram, 156 of a simplicial scheme, 109

motivic Borel-Moore functor, 216 Borel-Moore homology, 215 cohomology, 21, 22

and higher Chow groups, 103 and homology of GL, 121-122 as Zariski hypercohomology, 89 for simplicial schemes, 30-31 mod n, 21 of a diagram, 36 relative, 34, 123, 218, 225 semi-purity, 140 special properties, 89

cohomology with compact support, 215 cycle complex, 67 cycles functor, 38 DG category, 12, 36

construction, 12-16 structure, 36-38

functor with compact support, 216 Gersten complex, 91 Gersten conjecture, 92 Gersten resolution, 93 homological functor, 215 homology, 215

relative, 225 homotopy category, 40

structure, 40-44 hypercohomology, 62 local to global spectral sequence, 90 polylogarithm, 303

as a diagram of schemes, 303-304 as a motive, 309 distinguished triangle, 309 spectral sequence, 305-306

pull-back, 24 Quillen spectral sequence, 91 suspension, 68, 70, 76, 187 triangulated category, 16, 44

definition, 17-19 structure, 44-47

triangulated Tate category, 234, 235 moving lemma, 20, 94, 96, 102

for diagrams, 36 isomorphism, 18

multiplication commutative, 453 for a cosimplicial functor, 464 in a symmetric monoidal category, 453

SUBJECT INDEX 511

of cosimplicial objects, 452 multiplicative system, 424-425

N n-cubes, 32

distinguished triangle, 33 lifting, 32 relative motives, 31

negligible complexes, 270 nerve of a category, 468 Nisnevic sheaf with transfers, 311

homotopy invariant, 311 Nisnevic topology, 311 normal crossing subschemes, 208 normalized complex

of a cosimplicial abelian group, 275 of an inverse system, 269

O octahedra, 429 operad, 444

P pentagonal identity, 375 permutative bi-module, 402-403 perverse sheaf, 282 plus construction

and Q-construction, 359 for diagrams, 361-362

H-group structure, 365 for rings, 359 for schemes, 360

point, 483 associated pro-object, 485 conservative family, 484 neighborhood of a, 485 stalk, 484

pre-additive category, 378 pre-differential graded category, 379 pre-graded category, 378 presheaf

of sets, 482 with values in a category, 482

pre-tensor category, 382 Pre-Tr, 410-411 products

Alexander-Whitney, 452 and holim, 476-479 for cosimplicial objects, 452 for equi-dimensional cycles, 355 for motives of diagrams, 36 for motives of simplicial schemes, 31 for motivic Borel-Moore homology, 217 for motivic cohomology, 22 for motivic cohomology with compact sup­

port, 217 for motivic homology, 217

projection formula for a projection, 144

for a projective morphism, 150 for an embedding, 136

projective bundle formula, 113, 114 pseudo-tensor functor, 382-383 pseudo-abelian hull, 427

of a tensor category, 427 of a triangulated category, 427-433

push-forward additional properties, 151 for K-theory, 171-172 for a projection, 142 for a projective morphism, 146

compatibility with products, 151 compatibility with cycle classes, 152 compatibility with pull-back, 150 definition, 146-147 functoriality, 149 naturality, 151

for diagrams, 153, 159-161

Q Q-construction, 358

R real Frobenius, 268 realization

Betti, 267 complex, 267-268 real, 268

Chow, 79, 80 etale, 268, 271-272

Qi, 272 mod n, 272

for geometric cohomology theories, 261 functor, 260, 266

for cohomology, 267 Hodge, 273, 280, 281

over a smooth base, 282, 284 real, 281, 282

motivic, 284, 291 reduction, 240 relative cycles, 331 relativization

distinguished triangle, 33 sequence, 126, 129

compatibility with Chern classes, 128-130

resolution of singularities, 237 Riemann-Roch

for singular schemes, 231, 233 for smooth schemes, 176, 177 without denominators, 171, 174

S semi-monoidal category, 375-376

strictly associative, 375 sheaf

of .R-modules, 486 flat, 497

512 SUBJECT INDEX

of sets, 482 with values in a category, 482

sheafiflcation of a presheaf, 482 simplicial and cosimplicial objects, 449 simplicial closed subset, 109-110 simplicial schemes

lifting, 29 line bundles on, 112 vector bundles on, 110

smoothly decomposable scheme, 225 splitting idempotents, 433 splitting principle

for K0l 164 for the motive of a diagram, 161 for the motive of a simplicial scheme, 116

5-smooth stratification, 225 stabilization map, 358 stalk, 484

of a presheaf, 486 of a sheaf, 486

Steinberg relation, 295, 296 structured category, 377 support of a cycle, 332 suspension, 67 symmetric monoidal category, 375, 380

constructions, 383, 385, 386 examples, 376 graded, 436

symmetric monoidal V-category, 380 free, 384

symmetric semi-monoidal category, 375

T Tate mixed Hodge structure, 279 Tate twist, 20 tensor category, 381

DG, 382 without unit, 382

graded, 381 without unit, 382

triangulated, 416 thick subcategory, 424

generated by a set of objects, 425 Thom-Sullivan cochains, 275-277

tempered, 287-289 Todd character, 176 Todd class, 176 topos, 483

as a site, 483 fiber functor, 483 point of a, 483 with enough points, 484

total Chern class, 116 total complex functor, 412-413 translation structure, 403

for complexes over an additive category, 409

for Pre-Tr, 411 free, 406-409

free compatible, 408 tensor compatible, 404

translation structures, 401 transverse cartesian square, 150 triangulated category, 414

axioms, 414-415 exact functor, 415 five lemma, 415 localization, 424-426 multiplicative system, 424-425 of complexes over a DG category, 416 pseudo-abelianization, 427 thick subcategory, 424

triangulated tensor category, 416 localization, 426

triangulation, 477 twisted duality theory, 258 2-category, 379 2-resolution, 239

strict, 240 weak, 247

U unit isomorphism, 18 unit motive, 20 universal vector bundle, 358

V V-category, 377 V-functor, 377 V-natural transformation, 378

W Whitney product formula, 117

Z Z/2-set, 435

equivalence relations, 436 graded, 435

Index of Notation

S c h s , 9 S m 5 , 9 S m | s s , 9 £(V), 9 Zd{X/S), 10 Zd(X)f, 10 £(V)* ,11 ^ i ( V ) , 12 Z x ( d ) / , 12 ^2(V) , 12 ^s (V) , 13 >U(V), 14 ^ s (V) , 14 Anot(V), 15 ^S(V), 15 CLt(V), 16 KLt(V), 16 DLt(V), 17 PA1(V) H , 19 Z x ,x> 19 U[Z]vr, 20 U[|s(X)|]op*, 20 z | * ( 0 ) , 28 Zx(9)7~*,29 ^ n ( V ) * , 3 7 2 m o t , 40 CLt(V)*,40 K L t W * , 4 0 KLt(V)*B, 42 DLt(V)*,44 DLt(V) : d d , 45 c l x , x ' 4 8

H R , 59 Eg, 62 Zar(X), 63

(Zar (X, / ) ) , 63 C ^ o t ( Z a r ( X , / ) ) , 63 CH«(X,p),.66 E N r , 68 -2mot(r, *), 68 £mot(*), 69 CH n a z / ( r , p ) , 70 Ci-naifi ' 1

C H ^ X / S . p ) , 72 d"vp

f, 72

CH, 76 cl, 76 Z « ( X / 5 , * ) / , 77 c l^ p , 77 CHq(X/S,p), 77

3x/s(*)/' 7 7

^3mot(*), 80 »cw, 80 2mot ( r ,* ,* ) , 84 3X/s(*>*)/>84

^3X / S ( * ) ( * ) / , 85

ci(L), 112 c p (£ ) , 116 GCN, 121 BQCN/X, 121 Z B G L J V ( A ) , 121 c ^ " p , 123

q,2q-p 1 2 4

Z|M-, 153 *, 162 cx,p, 165 chx,P, 169 Todd, 176 Z«(X;Di,... ,D n), 181 t'(A,£), 192 i"(i4,B), 192 VM(VYT, 206 Dmot(V)pr, 207 Motfl(fe), 214 Sm5' |, 215

5,proj ' Sm^pro j , 215 Zh

x, 215 Hp(X,Z(q)), 215 //J*-M-(X,Z(g)), 215 Z ^ / 5 , 215 ff c

P/ 5(X,Z(g)) ,215

HP((X;DU... ,D n ) ,Z(g) ) , 225 Zf-M-, 225 S D S S , 227 SDSsproj, 227 SDSg r

p r o j ,229 VTM(S)R, 235

513

514 INDEX O F NOTATION

VT(S)R, 235 Sch£n , 237 • £ , 239 D n , 239 Sch£ p t , 251 Sch£n

p r , 251 Sh^ T , 256

Sh | ' | X (*) f l , 258 ^ ( V ) , 261 ^ (V), n = 3, 4, 5, mot, 263 X>MS ?(V)A ,267 S r 4 / r ( X ) , 269 D * l i m S h ^ ( X ) , 270 Q* <g> G, 276

r | , 2 7 7 Dec(W), 278 n*'«, 287 Q** ®G, 289

<— ^G m , 293 D 1 , 293 D n , 293 c l G ^ ) X , 2 9 4

K™(F), 298 /C^ , 303 ftn(Z(n)), 303 SmCor(k), 311 c(X,Y), 311 Sm/k, 311 L(X), 311 S/ijv»fl(SmCor(fc)), 311 DMeJS{k), 311 DMe

gfJ(k), 313

DMgm{k), 313 LC(X), 314 cpcf1,*^2), 314 3(Y,X), 320 P A A 325 X/f> 325 ^ n / A , 3 2 5 ^ q ( X ) , 326 VMh(k), 329 dim/^W), 331 C d (X/5) , 332 mK:fc(^7VK), 338 771(1^; W, s) (for a normal base), 340 Zd(X/S)z, 347 m(VT; W, s) (for a reduced base), 349 Zd(X/S\ 349 Zd(X/S), 349 EG, 357 BG, 357 G*(G;A), 357 GLAT(A) , 358 GLiv/5 , 358 BQM, 358

K p , 358 K0 , 359 M x , 359 P x , 359 BGLiv(A)+ , 359 [n], 360 K(X,U), 360 7>x,w, 360 £ £ £ , 363 M o d A , 376 G r M o d A , 376 D G - M o d A , 376 c a t v , 378 w, 387 &n-^mi oo7 CJO, 387 5 < _ m , 387 too, 387 ft, 389 C®'c, 391 M[n], 401 M W , 401 A[*], 406 -4[*j®, 407 C 6 ( ^ ) , 411 Kb(A), 411 Tot, 412 Z/2-Sets, 435 E, 440 IM**, 442 C ® ^ 446 A, 449 [n], 449 <5™, 449 erf1, 449 c.s.C, 449 s.C, 449 A ^ n , 449 An.d., 449 A n . d . / [n] , 449 Z®(F*),450 Z n (F*) , 450 6 ^ ' n , 450 X ^ , 450 A u n , 455 C®'sf>, 461 Ksf>, 463 ccS®'sf>, 465 holim/, 468 G*(5;Z), 469 G*(5;Z) n . d . ,469 I/i, 469 holim/?n.d., 471 sgn(s), 477 Cov(X), 481 Top, 481 Sch^ t , 482 Shf C j T ) ,482

INDEX OF NOTATION 515

C, 482 C, 482 Point (T), 483 CRl 486

Selected Titles in This Series (Continued from the front of this publication)

25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W . Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe , Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 21 Albert Baernste in , David Drasin, Peter Duren, and Albert Marden, Editors,

The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986

20 Kenneth R. Goodearl , Partially ordered abelian groups with interpolation, 1986 19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, 1984 18 Prank B. Knight, Essentials of Brownian motion and diffusion, 1981 17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, 1980 16 O. T imothy O'Meara, Symplectic groups, 1978 15 J. Dieste l and J. J. Uhl , Jr., Vector measures, 1977 14 V. Guil lemin and S. Sternberg, Geometric asymptotics, 1977 13 C. Pearcy, Editor, Topics in operator theory, 1974 12 J. R. Isbell, Uniform spaces, 1964 11 J. Cronin, Fixed points and topological degree in nonlinear analysis, 1964 10 R. Ayoub, An introduction to the analytic theory of numbers, 1963 9 Arthur Sard, Linear approximation, 1963 8 J. Lehner, Discontinuous groups and automorphic functions, 1964

7.2 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume II, 1961 7.1 A. H. Clifford and G. B. Pres ton , The algebraic theory of semigroups, Volume I, 1961

6 C. C. Chevalley, Introduction to the theory of algebraic functions of one variable, 1951 5 S. Bergman, The kernel function and conformal mapping, 1950 4 O. F. G. Schilling, The theory of valuations, 1950 3 M. Marden, Geometry of polynomials, 1949 2 N . Jacobson, The theory of rings, 1943 1 J. A. Shohat and J. D . Tamarkin, The problem of moments, 1943