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

We show that $\bigoplus_{n \ge 0} {\mathrm H}^t({\bf GL}_n({\bf F}q), {\bf F}\ell)$canonicallyadmitsthestructureofamoduleoverthe$q$-divided power algebra (assuming $q$ is invertible in ${\mathbf{F}}_{\mathrm{\ell }}$), and that, as such, it is free and (for $q\mathrm{\ne }2$) generated in degrees $\le t$. As a corollary, we show that the cohomology of a finitely generated ${\bf VI}$-moduleinnon-describingcharacteristiciseventuallyperiodicin$n$. We apply this to obtain a new result on the cohomology of unipotent Specht modules.