Lecture Notes on Motivic Cohomology (Clay Mathematics Monographs) (Clay Mathematics Monographs)

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). 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 The notion of a motive is an elusive one, like its namesake 'the motif' of Cezanne's impressionist method of painting. This title includes lectures that correspond to one-hour lectures given by the author during the course he gave at the Institute for Advanced Study in Princeton on this subject in 1999-2000.