The Gm-equivariant Motivic Cohomology of Stiefel Varieties

TL;DR

This paper develops a spectral sequence for cohomology theories in simplicial presheaves and applies it to compute the Gm-equivariant motivic cohomology of Stiefel varieties, linking it to equivariant higher Chow groups.

Contribution

It introduces a new spectral sequence framework and calculates the Gm-equivariant motivic cohomology of Stiefel varieties, expanding computational tools in motivic homotopy theory.

Findings

Derived a spectral sequence for cohomology in simplicial presheaves.

Computed Gm-equivariant motivic cohomology of Stiefel varieties.

Connected equivariant motivic cohomology with higher Chow groups.

Abstract

We derive a version of the Rothenberg-Steenrod, fiber-to-base, spectral sequence for cohomology theories represented in model categories of simplicial presheaves. We then apply this spectral sequence to calculate the equivariant motivic cohomology of the general linear group with a general Gm-action, this coincides with the equivariant higher Chow groups. Some of the equivariant motivic cohomology of a Stiefel variety with a general Gm-action is deduced as a corollary.