About Bredon motivic cohomology of a field

TL;DR

This paper establishes a method to compute Bredon motivic cohomology over perfect fields using Suslin-Friedlander complexes, and identifies it explicitly for certain types of fields in specific weights.

Contribution

It introduces a new computational approach for Bredon motivic cohomology and explicitly determines it for quadratically closed and Euclidean fields in weights 1 and σ.

Findings

Bredon motivic cohomology can be computed via Suslin-Friedlander complexes.

Explicit identification of Bredon motivic cohomology for quadratically closed and Euclidean fields.

Bredon motivic cohomology in weight 0 with integer coefficients matches Bredon cohomology of a point.

Abstract

We prove that, over a perfect field, Bredon motivic cohomology can be computed by Suslin-Friedlander complexes of equivariant equidimensional cycles. Partly based on this result we completely identify Bredon motivic cohomology of a quadratically closed field and of a euclidian field in weights 1 and $\sigma$. We also prove that Bredon motivic cohomology of an arbitrary field in weight 0 with integer coefficients coincides (as abstract groups) with Bredon cohomology of a point.