Lecture Notes on Motivic Cohomology

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American Mathematical Soc., Jan 24, 2011 - Mathematics - 216 pages
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The notion of a motive is an elusive one, like its namesake "the motif" of Cezanne's impressionist method of painting. Its existence was first suggested by Grothendieck in 1964 as the underlying structure behind the myriad cohomology theories in Algebraic Geometry. We now know that there is a triangulated theory of motives, discovered by Vladimir Voevodsky, which suffices for the development of a satisfactory Motivic Cohomology theory. However, the existence of motives themselves remains conjectural. The lecture notes format is designed for the book to be read by an advanced graduate student or an expert in a related field. The lectures roughly correspond to one-hour lectures given by Voevodsky during the course he gave at the Institute for Advanced Study in Princeton on this subject in 1999-2000. In addition, many of the original proofs have been simplified and improved so that this book will also be a useful tool for research mathematicians. This book provides an account of the triangulated theory of motives. Its purpose is to introduce Motivic Cohomology, to develop its main properties, and finally to relate it to other known invariants of algebraic varieties and rings such as Milnor K-theory, etale cohomology, and Chow groups. The book is divided into lectures, grouped in six parts. The first part presents the definition of Motivic Cohomology, based upon the notion of presheaves with transfers. Some elementary comparison theorems are given in this part. The theory of (etale, Nisnevich, and Zariski) sheaves with transfers is developed in parts two, three, and six, respectively. The theoretical core of the book is the fourth part, presenting the triangulated category of motives. Finally, the comparison with higher Chow groups is developed in part five.
 

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Contents

The category of finite correspondences
3
Presheaves with transfers
13
Motivic cohomology
21
Relation to Milnor KTheory
29
Etale sheaves with transfers
37
The relative Picard group and Suslins Rigidity Theorem
47
Derived tensor products
55
Appendix 8A Tensor triangulated categories
63
Equidimensional cycles
125
Motives with compact support
128
Part5 Higher Chow Groups
133
Higher Chow groups
135
Appendix 17A Cycle maps
143
Higher Chow groups and equidimensional cycles
149
Appendix 18A Generic equidimensionality
155
Motivic cohomology and higher Chow groups
159

Etale motivic cohomology and algebraic singular homology
75
Standard triples
83
Nisnevich sheaves
89
Nisnevich sheaves with transfers
99
Cdh sheaves with transfers
105
The Triangulated Category of Motives
107
The category of motives
109
Nisnevich A1local complexes
111
Motives with Qcoefficients
116
The complex Zn and Pn
119
Geometric motives
167
Zariski Sheaves with Transfers
173
Covering morphisms of triples
175
Zariski sheaves with transfers
183
Contractions
191
Homotopy invariance of cohomology
197
Bibliography
203
Glossary
207
Index
211
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