Annals of Mathematics Studies Number 143 Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:30 PM Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:30 PM Cycles, Transfers, and Motivic Homology Theories by Vladimir Voevodsky, Andrei Suslin, and Eric M. Friedlander PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 2000 Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:30 PM Copyright © 2000 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved The Annals of Mathematics Studies are edited by John N, Mather and Elias M. Stein Library of Congress Catalog Card Number 00-100291 ISBN 0-691-04814-2 (cloth) ISBN 0-691-04815-0 (pbk.) The publisher would like to acknowledge the authors for providing the electronic files that were reformatted to produce the camera-ready copy from which this book was printed The paper used in this publication meets the minimum requirements of ANSI/NISO Z39.48-1992 (R 1997) (Permanence of Paper) www.pup.princeton.edu Printed in the United States of America 1 3 5 79 10 8 6 42 3 5 79 10 8 6 42 (Pbk.) Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:30 PM Contents Chapter 1 Introduction 3 Eric M. Friedlander, A. Suslin, and V. Voevodsky Chapter 2 Relative Cycles and Chow Sheaves 10 Andrei Suslin and Vladimir Voevodsky Chapter 3 Cohomological Theory of Presheaves with Transfers 87 Vladimir Voevodsky Chapter 4 Bivariant Cycle Cohomology 138 Eric M. Friedlander and Vladimir Voevodsky Chapter 5 Triangulated Categories of Motives Over a Field 188 Vladimir Voevodsky Chapter 6 Higher Chow Groups and Etale Cohomology 239 Andrei A. Suslin v Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:30 PM Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:30 PM Cycles, Transfers, and Motivic Homology Theories Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:30 PM Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:30 PM 1 Introduction Eric M. Friedlander, A. Suslin, and V. Voevodsky Our original goal which finally led to this volume was the construction of "motivic cohomology theory," whose existence was conjectured by A. Beilinson and S. Lichtenbaum ([2], [3], [17], [18]). Even though this would seem to be achieved at the end of the third paper, our motivation evolved into a quest for a deeper understanding of various properties of algebraic cycles. Thus, several of the papers presented here do not deal directly with motivic cohomology but rather with basic questions about algebraic cycles. In this introduction, we shall begin with a short reminder of A. Beilin- son's formulation of motivic cohomology theory. We then proceed to briefly summarize the topic and contents of individual papers in the volume. Let k be a field and Sm/k denote the category of smooth schemes over k. Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:22 PM 10 Chapter 1 — E. M. Friedlander, A. Suslin, and V. Voevodsky This definition of motivic cohomology is not "topology free." In par- ticular one may consider the corresponding hypercohomology groups in the etale topology instead of the Zariski topology. S. Lichtenbaum ([17], [18]) has in fact suggested axioms for the etale analog of Beilinson's mo- tivic cohomology. We emphasize that everywhere in this volume "motivic (co-)homology" mean motivic (co-)homology in the Zariski topology unless the etale topology is explicitly specified. In addition to the axioms given above, Beilinson's original list contained two further axioms. These we state below in the form of conjectures. Beilinson-Lichtenbaum Conjecture. For a field F over k and a prime I not equal to char(k), one has In conjunction with the spectral sequence relating motivic cohomology to algebraic K-theory, these two "axioms" imply the validity of highly non- trivial conjectures in algebraic K-theory. Brought to you by | The Royal Library (Det Kongelige Bibliotek) - National Library of Denmark / Copenhagen University Library Authenticated Download Date | 11/20/19 4:22 PM