Notes on Motivic Cohomology

Carlo Mazza, Vladimir Voevodsky, Charles Weibel9780821838471, 0821838474

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, étale 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 (étale, 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.
Titles in this series are copublished with the Clay Mathematics Institute (Cambridge, MA).

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Table of contents :
The category of finite correspondences……Page 11
The category CorS……Page 17
Presheaves with transfers……Page 23
Motivic cohomology……Page 33
Weight one motivic cohomology……Page 39
Relation to Milnor K-Theory……Page 43
Étale sheaves with transfers……Page 51
Relative Picard group and ………Page 63
Derived tensor products……Page 73
Tensor Triangulated Categories……Page 83
A1-weak equivalence……Page 87
Étale motivic cohomology and ………Page 97
Standard triples……Page 105
Nisnevich sheaves……Page 113
Nisnevich sheaves with transfers……Page 121
The category of motives……Page 127
The complex Z(n) and Pn……Page 135
Equidimensional cycles……Page 141
Higher Chow groups……Page 147
Cycle maps……Page 157
Higher Chow groups and ………Page 163
Generic Equidimensionality……Page 171
Motivic cohomology and ………Page 175
Covering morphisms of triples……Page 183
Zariski sheaves with transfers……Page 193
Contractions……Page 203
Homotopy Invariance of Cohomology……Page 209