The etale cohomology of the general linear group over a finite field and the Deligne and Lusztig variety

TL;DR

This paper investigates the etale cohomology of the general linear group over finite fields using stratification techniques and Deligne-Lusztig varieties, providing new computational insights into these algebraic structures.

Contribution

It introduces a novel approach combining stratification methods with Deligne-Lusztig varieties to compute etale cohomology of GL_n over finite fields.

Findings

Computed etale cohomology groups for specific cases

Established connections between stratification and cohomology

Enhanced understanding of the structure of Deligne-Lusztig varieties

Abstract

Let $p\not =\ell$ be primes. We study the etale cohomology $H^{*}_{et}(BGL_n(\bF_{p^s});\bZ/{\ell})$ over the algebraically closed field $\bar \bF_p$ by using the stratification methods from Molina-Vistoli. To compute this cohomology, we use the Delinge-Lusztig variety.