On the mod $p$ unramified cohomology of varieties having universally trivial Chow group of zero-cycles

TL;DR

This paper extends previous results on the triviality of the Brauer group for varieties with trivial Chow groups to unramified mod p étale motivic cohomology, removing the properness condition using Suslin homology.

Contribution

It generalizes the triviality results from the Brauer group to broader unramified motivic cohomology groups and removes the properness assumption.

Findings

Universal triviality of unramified mod p étale motivic cohomology groups for certain varieties.

Removal of properness assumption using Suslin homology.

Extension of known results to a wider class of cohomology groups.

Abstract

Auel-Bigazzi-B\"ohning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as well. In this paper, we generalize their argument to arbitrary unramified mod $p$ \'etale motivic cohomology groups. We also see that the properness assumption on the variety $X$ can be dropped off by using the Suslin homology together with a certain tame subgroup of the unramified cohomology group.