Lecture Notes on Motivic Cohomology
Published on 30. January 2011
Book
Paperback/Softback
216 pages
978-0-8218-5321-4 (ISBN)
Description
A co-publication of the AMS and Clay Mathematics Institute
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.
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.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
College/higher education
Professional and scholarly
Weight
440 gr
ISBN-13
978-0-8218-5321-4 (9780821853214)
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Schweitzer Classification
Persons
Carlo Mazza, Rutgers University, Piscataway, NJ.
Vladimir Voevodsky, Institute for Advanced Study, Princeton, NJ.
Charles Weibel, Rutgers University, New Brunswick, NJ.
Vladimir Voevodsky, Institute for Advanced Study, Princeton, NJ.
Charles Weibel, Rutgers University, New Brunswick, NJ.
Content
* Etale motivic theory: * Etale sheaves with transfers * The relative Picard group and Suslin's rigidity theorem * Derived tensor products $\mathbb{A}^1$-weak equivalence * Etale motivic cohomology and algebraic singular homology
* Nisnevich sheaves with transfers: * Standard triples * Nisnevich sheaves* * Nisnevich sheaves with transfers
* The triangulated category of motives: * The category of motives * The complex $\mathbb{Z}(n)$ and $\mathbb{P}^n$ Equidimensional cycles
* Higher Chow groups: * Higher Chow groups * Higher Chow groups and equidimensional cycles * Motivic cohomology and higher Chow groups * Geometric motives
* Zariski sheaves with transfers: Covering morphisms of triples * Zariski sheaves with transfers * Contractions * Homotopy invariance of cohomology * Bibliography * Glossary * Index
* Nisnevich sheaves with transfers: * Standard triples * Nisnevich sheaves* * Nisnevich sheaves with transfers
* The triangulated category of motives: * The category of motives * The complex $\mathbb{Z}(n)$ and $\mathbb{P}^n$ Equidimensional cycles
* Higher Chow groups: * Higher Chow groups * Higher Chow groups and equidimensional cycles * Motivic cohomology and higher Chow groups * Geometric motives
* Zariski sheaves with transfers: Covering morphisms of triples * Zariski sheaves with transfers * Contractions * Homotopy invariance of cohomology * Bibliography * Glossary * Index