The $\mathbb{G}_m$–equivariant motivic cohomology of Stiefel varieties

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 ${GL}_n$ with a general ${\mathbb{G}}_m$–action; this coincides with the equivariant higher Chow groups. The motivic cohomology of ${PGL}_n$ and some of the equivariant motivic cohomology of a Stiefel variety, $V_m\left({\mathbb{A}}^n\right)$, with a general ${\mathbb{G}}_m$–action is deduced as a corollary.