Periodicity in the cohomology of finite general linear groups via q-divided powers

Abstract

We show that ${⨁}_{n\ge 0}{\mathrm{H}}^t({\mathbf{GL}}_n({\mathbf{F}}_q),{\mathbf{F}}_{\ell })$ canonically admits the structure of a module over the $q$-divided power algebra (assuming $q$ is invertible in ${\mathbf{F}}_{\ell }$), and that, as such, it is free and (for $q\ne 2$) generated in degrees $\le t$. As a corollary, we show that the cohomology of a finitely generated $\mathbf{VI}$-module in non-describing characteristic is eventually periodic in $n$. We apply this to obtain a new result on the cohomology of unipotent Specht modules.

19 pages

References in corpus (5)

• FI-modules and the cohomology of modular representations of symmetric groups
• Periodicity in the cohomology of symmetric groups via divided powers
• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• Homological stability for classical groups
• Modular characteristic classes for representations over finite fields