Divisibility properties of motivic cohomology

TL;DR

This paper investigates the divisibility properties of motivic cohomology, extending known results to a broader class of varieties and deriving implications like spectral sequence degeneration.

Contribution

It generalizes previous results on $ ext{K}_2$-cohomology to étale motivic cohomology for non-projective varieties over separably closed fields.

Findings

Degeneration of the Bloch-Lichtenbaum spectral sequence for any field containing $k$

Extension of divisibility results to non-projective varieties

Broader understanding of motivic cohomology properties

Abstract

We extend results of Colliot-Th\'el\`ene and Raskind on the $\mathcal{K}_2$-cohomology of smooth projective varieties over a separably closed field $k$ to the \'etale motivic cohomology of smooth, not necessarily projective, varieties over $k$. Some consequences are drawn, such as the degeneration of the Bloch-Lichtenbaum spectral sequence for any field containing $k$.