Motivic Cohomology

Prerequisite

Basic algebraic geometry (GTM 52, Chapter 1-3)

Abstract

Motivic cohomology, originated from Deligne, Beilinson and Lichtenbaum and developed by Voevodsky, is a kind of cohomology theory on schemes. It admits comparison with étale cohomology of powers of roots of unity (Beilinson-Lichtenbaum), together with higher Chow groups, and relates to K-theory by Atiyah-Hirzebruch spectral sequence. In this lecture, we establish the category of motives in which the motivic cohomologies are realized. We explain its relationship with Milnor K-theory and Chow group, as well as the theory of cycle modules. Furthermore, we introduce cancellation theorem, Gysin triangle, projective bundle formula, BB-decomposition and duality.

Lecturer Intro.

Nanjun Yang got his doctor and master degree in University of Grenoble-Alpes, advised by Jean Fasel, and bachelor degree in Beihang University. Currently he is a assistant researcher in BIMSA. His research interests are motivic cohomology and Chow-Witt ring. He proposed the theory of split Milnor-Witt motives, which applies to the computation of the Chow-Witt ring of fiber bundles. The corresponding results have been published independently on journals such as Camb. J. Math and Doc. Math.