# Lecture Notes on Motivic Cohomology

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*Carlo Mazza; Vladimir Voevodsky; Charles Weibel*

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,
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 co-published with the Clay Mathematics Institute (Cambridge, MA).

#### Readership

Graduate students and research mathematicians interested in algebraic geometry.