#### Volume 13, issue 2 (2013)

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

### Ben Williams

Algebraic & Geometric Topology 13 (2013) 747–793
##### 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.

##### Keywords
Equivariant, Motivic cohomology, Chow group, Fiber-to-base, Stiefel, Projective general linear group
##### Mathematical Subject Classification 2010
Primary: 19E15
Secondary: 14C15, 18G55
##### Publication
Received: 28 March 2012
Revised: 2 August 2012
Accepted: 6 August 2012
Published: 24 March 2013
##### Authors
 Ben Williams Department of Mathematics University of Southern California Kaprielian Hall 3620 South Vermont Avenue Los Angeles, CA 90089-2532 USA