FI-modules and the cohomology of modular representations of symmetric groups

Abstract: An FI-module $V$ over a commutative ring $\bf{k}$ encodes a sequence $(V_n){n \geq 0}$ of representations of the symmetric groups $(\mathfrak{S}_n){n \geq 0}$ over $\bf{k}$. In this paper, we show that for a "finitely generated" FI-module $V$ over a field of characteristic $p$, the cohomology groups $H^t(\mathfrak{S}_n, V_n)$ are eventually periodic in $n$. We describe a recursive way to calculate the period and the periodicity range and show that the period is always a power of $p$. As an application, we show that if $\mathcal{M}$ is a compact, connected, oriented manifold of dimension $\geq 2$ and $\mathit{conf}_n(\mathcal{M})$ is the configuration space of unordered $n$-tuples of distinct points in $\mathcal{M}$ then the mod-$p$ cohomology groups $H^{t}(\mathit{conf}_n(\mathcal{M}),\bf{k})$ are eventually periodic in $n$ with period a power of $p$.