  We show that $\bigoplus_{n \ge 0} {\mathrm H}^t({\bf GL}_n({\bf F}_q), {\bf
F}_\ell)$ canonically admits the structure of a module over the $q$-divided
power algebra (assuming $q$ is invertible in ${\bf F}_{\ell}$), and that, as
such, it is free and (for $q \neq 2$) generated in degrees $\le t$. As a
corollary, we show that the cohomology of a finitely generated ${\bf
VI}$-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.
