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Lecture Notes on Motivic Cohomology Lecture Notes on Motivic Cohomology Clay Mathematics Monographs Volume 2 Lecture Notes on Motivic Cohomology Carlo Mazza Vladimir Voevodsky Charles Weibel American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 14F42; Secondary 19E15, 14C25, 14–01, 14F20. For additional information and updates on this book, visit www.ams.org/bookpages/cmim-2 Library of Congress Cataloging-in-Publication Data Mazza, Carlo, 1974– Lecture notes on motivic cohomology / Carlo Mazza, Vladimir Voevodsky, Charles A. Weibel. p. cm. — (Clay mathematics monographs, ISSN 1539-6061 ; v. 2) Includes bibliographical references and index. ISBN-10: 0-8218-3847-4 (alk. paper) ISBN-13: 978-0-8218-3847-1 (alk. paper) 1. Homology theory. I. Voevodsky, Vladimir. II. Weibel, Charles A., 1950– III. Title. IV. Series. QA612.3.M39 2006 514.23—dc22 2006045973 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 is permitted only under license from the Clay Mathematics Institute. Requests for such permission should be addressed to the Clay Mathematics Institute, One Bow Street, Cambridge, MA 02138, USA. Requests can also be made by e-mail to permissions@claymath.org. c 2006 by the authors. All rights reserved. Published by the American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA. 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 http://www.ams.org/ Visit the Clay Mathematics Institute home page at http://www.claymath.org/ 10987654321 111009080706 Contents Preface vii Introduction xi Part 1. Presheaves with Transfers 1 Lecture 1. The category of finite correspondences 3 Appendix 1A. The category CorS 7 Lecture 2. Presheaves with transfers 13 Homotopy invariant presheaves 17 Lecture 3. Motivic cohomology 21 Lecture 4. Weight one motivic cohomology 25 Lecture 5. Relation to Milnor K-Theory 29 Part 2. Etale´ Motivic Theory 35 Lecture 6. Etale´ sheaves with transfers 37 Lecture 7. The relative Picard group and Suslin’s Rigidity Theorem 47 Lecture 8. Derived tensor products 55 Appendix 8A. Tensor triangulated categories 63 Lecture 9. A1-weak equivalence 67 Etale´ A1-local complexes 71 Lecture 10. Etale´ motivic cohomology and algebraic singular homology 75 Part 3. Nisnevich Sheaves with Transfers 81 Lecture 11. Standard triples 83 Lecture 12. Nisnevich sheaves 89 The cdh topology 94 v vi CONTENTS Lecture 13. Nisnevich sheaves with transfers 99 Cdh sheaves with transfers 105 Part 4. The Triangulated Category of Motives 107 Lecture 14. The category of motives 109 Nisnevich A1-local complexes 111 Motives with Q-coefficients 116 Lecture 15. The complex Z(n) and Pn 119 Lecture 16. Equidimensional cycles 125 Motives with compact support 128 Part 5. Higher Chow Groups 133 Lecture 17. Higher Chow groups 135 Appendix 17A. Cycle maps 143 Lecture 18. Higher Chow groups and equidimensional cycles 149 Appendix 18A. Generic equidimensionality 155 Lecture 19. Motivic cohomology and higher Chow groups 159 Lecture 20. Geometric motives 167 Part 6. Zariski Sheaves with Transfers 173 Lecture 21. Covering morphisms of triples 175 Lecture 22. Zariski sheaves with transfers 183 Lecture 23. Contractions 191 Lecture 24. Homotopy invariance of cohomology 197 Bibliography 203 Glossary 207 Index 211 Preface This book was written by Carlo Mazza and Charles Weibel on the basis of the lectures on motivic cohomology which I gave at the Institute for Advanced Study in Princeton in 1999/2000. From the point of view taken in these lectures, motivic cohomology with coef- ficients in an abelian group A is a family of contravariant functors H p,q(−,A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by integers p and q. The idea of motivic cohomology goes back to P. Deligne, A. Beilinson and S. Lichtenbaum. Most of the known and expected properties of motivic cohomology (predicted in [ABS87]and[Lic84]) can be divided into two families. The first family concerns properties of motivic cohomology itself – there are theorems about homotopy in- variance, Mayer-Vietoris and Gysin long exact sequences, projective bundles, etc. This family also contains conjectures such as the Beilinson-Soule´ vanishing con- jecture (H p,q = 0forp < 0) and the Beilinson-Lichtenbaum conjecture, which can be interpreted as a partial etale´ descent property for motivic cohomology. The sec- ond family of properties relates motivic cohomology to other known invariants of algebraic varieties and rings. The power of motivic cohomology as a tool for prov- ing results in algebra and algebraic geometry lies in the interaction of the results in these two families; applying general theorems of motivic cohomology to the specific cases of classical invariants, one gets new results about these invariants. The idea of these lectures was to define motivic cohomology and to give care- ful proofs for the elementary results in the second family. In this sense they are complementary to the study of [VSF00], where the emphasis is on the properties of motivic cohomology itself. The structure of the proofs forces us to deal with the main properties of motivic cohomology as well (such as homotopy invariance). As a result, these lectures cover a considerable portion of the material of [VSF00], but from a different point of view. One can distinguish the following “elementary” comparison results for motivic cohomology. Unless otherwise specified, all schemes below are assumed to be smooth or (in the case of local or semilocal schemes) limits of smooth schemes. vii viii PREFACE (1) H p,q(X,A)=0forq < 0, and for a connected X one has A for p = 0 H p,0(X,A)= 0forp = 0; (2) one has O∗(X) for p = 1 p,1 Z H (X, )= Pic(X) for p = 2 0forp = 1,2; p,p M ⊗ M (3) for a field k, one has H (Spec(k),A)=Kp (k) A where Kp (k) is the p-th Milnor K-group of k (see [Mil70]); (4) for a strictly Hensel local scheme S over k and an integer n prime to char(k), one has ⊗ µ q(S) for p = 0 H p,q(S,Z/n)= n 0forp = 0 where µn(S) is the groups of n-th roots of unity in S; (5) one has H p,q(X,A)=CHq(X,2q − p;A).HereCHi(X, j;A) denotes the higher Chow groups of X introduced by S. Bloch in [Blo86], [Blo94]. In particular, H2q,q(X,A)=CHq(X) ⊗ A, where CHq(X) is the classical Chow group of cycles of codimension q modulo rational equivalence. The isomorphism between motivic cohomology and higher Chow groups leads to connections between motivic cohomology and algebraic K-theory, but we do not discuss these connections in the present lectures. See [Blo94], [BL94], [FS02], [Lev98]and[SV00]. Deeper comparison results include the theorem of M. Levine comparing CHi(X, j;Q) with the graded pieces of the gamma filtration in K∗(X)⊗Q [Lev94], and the construction of the spectral sequence relating motivic cohomology and al- gebraic K-theory for arbitrary coefficients in [BL94]and[FS02]. The lectures in this book may be divided into two parts, corresponding to the fall and spring terms. The fall term lectures contain the definition of motivic co- homology and the proofs for all of the comparison results listed above except the last one. The spring term lectures include more advanced results in the theory of sheaves with transfers and the proof of the final comparison result (5). The definition of motivic cohomology used here goes back to the work of An- drei Suslin in about 1985. As I understand it, when he came up with this definition he was able to prove the first three of the comparison results stated above. In par- ticular, the proof of comparison (3) between motivic cohomology and Milnor’s K-groups given in these lectures is exactly Suslin’s original proof. The proofs of the last two comparison results, (4) and (5), are also based on results of Suslin. Suslin’s formulation of the Rigidity Theorem ([Sus83]; see theorem 7.20) is a key PREFACE ix result needed for the proof of (4), and Suslin’s moving lemma (theorem 18A.1 below) is a key result needed for the proof of (5). It took ten years and two main new ideas to finish the proofs of the compar- isons (4) and (5). The first one, which originated in the context of the qfh-topology and was later transferred to sheaves with transfers (definition 2.1), is that the sheaf of finite cycles Ztr(X) is the free object generated by X. This idea led to a group of results, the most important of which is lemma 6.23. The second idea, which is the main result of [CohTh], is represented here by theorem 13.8. Taken together they allow one to efficiently do homotopy theory in the category of sheaves with transfers. A considerable part of the first half of the lectures is occupied by the proof of (4). Instead of stating it in the form used above, we prove a more detailed theorem. For a given weight q, the motivic cohomology groups H p,q(X,A) are defined as the hypercohomology (in the Zariski topology) of X with coefficients in a complex | of sheaves A(q) XZar .
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