Motivic cohomology and infinitesimal group schemes

TL;DR

This paper computes the mod p motivic cohomology of classifying spaces for infinitesimal group schemes over perfect fields, introducing a motivic Eilenberg-Moore spectral sequence and cycle class maps to étale and Hodge cohomology.

Contribution

It introduces a motivic Eilenberg-Moore spectral sequence and defines cycle class maps for classifying spaces of infinitesimal group schemes, expanding computational tools in motivic cohomology.

Findings

Computed motivic cohomology for Frobenius kernels of split reductive groups.

Established a cycle class map from motivic to étale and Hodge cohomology.

Analyzed examples including Frobenius kernels.

Abstract

For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ \'etale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels.