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Mixed Motives, 1998 56 Leonid I http://dx.doi.org/10.1090/surv/057 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 William 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 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam 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. Nation, 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 Guy 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 Nathan 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) Mathematical Surveys and Monographs Volume 57 Mixed Motives Marc Levine glEMgg. American Mathematical Society Editorial Board Georgia Benkart Tudor Stefan 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 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 X-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-permissionOams.org. © 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/ 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 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 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 cate­ gory 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 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
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