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

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