During the academic year 1999-2000, Voevodsky gave a course on motivic cohomology at the Institute for Advanced Study in Princeton. These lecture notes reflect the content of this course. They may be divided into two terms. The Fall term (Lectures 1-10) contains the basic definitions, organized around the notion of a presheaf with transfers, together with the fundamental comparisons with other known invariants: Picard group, Milnor K-theory and etale cohomology.
The Spring Term centers around Nisnevich sheaves with transfer. Lectures 11-14 contains the construction of the triangulated category of motives over a field, DM. The key technical result, that the cohomology of a homotopy invariant Nisnevich sheaf with transfers is homotopy invariant, is postponed to Lectures 21-23. Lectures 16-20 establish the isomorphism between motivic cohomology and higher Chow groups, without assuming resolution of singularities.
Here is the "Beta.1" version of the notes (221 pages, Fall 2003).
It replaces the Beta.0 version, which was posted July 2002.
Only minor changes were made between these versions.
ps file (1.4 MB)
pdf file (1.3 MB)
dvi file (0.77 MB) (Dvi file omits eps figures on pp.184, 189)
These lectures are a part of the Motivic Homotopy Theory Program