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

An FI-module $V$ over a commutative ring $\mathbf{k}$ encodes a sequence $(V_n{)}_{n\ge 0}$ of representations of the symmetric groups $({\mathfrak{S}}_n{)}_{n\ge 0}$ over $\mathbf{k}$. 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$$conf$_n($\mathcal{M}$)$istheconfigurationspaceofunordered$n$-tuplesofdistinctpointsin$$\mathcal{M}$$thenthemod-$p$ cohomology groups $H^t({conf}_n(\mathcal{M}),\mathbf{k})$ are eventually periodic in $n$ with period a power of $p$.