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

TL;DR

This paper proves that the cohomology groups of finitely generated FI-modules over fields of characteristic p exhibit eventual periodicity in n, with periods that are powers of p, and applies this to configuration spaces of manifolds.

Contribution

It establishes the eventual periodicity of cohomology groups for FI-modules over characteristic p fields and provides a recursive method to determine the period and range.

Findings

Cohomology groups are eventually periodic in n with period a power of p.

Provides a recursive method to compute the period and periodicity range.

Application to configuration spaces shows similar periodicity in manifold cohomology.

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 $p$.