Cycles, Transfers And Motivic Homology Theories

260 Pages·2000·9.506 MB·English

by  Vladimir Voevodsky, Eric M. Friedlander

Annals of Mathematics Studies
Number 143
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Cycles, Transfers, and
Motivic Homology Theories
by
Vladimir Voevodsky, Andrei Suslin,
and Eric M. Friedlander
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY
2000
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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
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ANSI/NISO Z39.48-1992 (R 1997) (Permanence of Paper)
www.pup.princeton.edu
Printed in the United States of America
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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
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Cycles, Transfers, and
Motivic Homology Theories
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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.
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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.
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