This book by C. Mazza and Ch. Weibel is based on a series of lectures given by V. Voevodsky in Princeton in the academic year 1999/2000. The idea of these lectures is to relate motivic cohomology to other known invariants of algebraic varieties and rings. The power of motivic cohomology as a tool for proving results in algebra and algebraic geometry lies in the interaction with properties of motivic cohomology itself – its homotopy invariance, Mayer-Vietoris and Gysin long exact sequences, projective bundles, etc. The contents of the book may be divided into two parts, corresponding to the fall and spring terms. The fall term lectures contain the definition of motivic cohomology and proofs for various comparison results (e.g. with the Milnor K-group and the Picard group). The spring term lectures include more advanced results in the theory of sheaves with transfers and a proof of the general comparison result with higher Chow groups of algebraic varieties.

Reviewer:

pso