# Periodicity in the cohomology of symmetric groups via divided powers

## TL;DR

This paper refines the understanding of cohomology periodicity in symmetric groups by showing that cohomology groups form a nearly finitely presented divided power module, leading to sharper bounds and applications in topology and representation theory.

## Contribution

It introduces a $ ext{D}$-module structure on cohomology of $ ext{FI}$-modules, generalizing stability to periodicity and providing new bounds and applications.

## Key findings

- Cohomology groups form a nearly finitely presented $ ext{D}$-module.
- Sharp bounds on the period of cohomology in characteristic zero.
- Applications to modular cohomology of Specht modules and configuration spaces.

## Abstract

A famous theorem of Nakaoka asserts that the cohomology of the symmetric group stabilizes. The first author generalized this theorem to non-trivial coefficient systems, in the form of $\mathrm{FI}$-modules over a field, though one now obtains periodicity of the cohomology instead of stability. In this paper, we further refine these results. Our main theorem states that if $M$ is a finitely generated $\mathrm{FI}$-module over a noetherian ring $\mathbf{k}$ then $\bigoplus_{n \ge 0} \mathrm{H}^t(S_n, M_n)$ admits the structure of a $\mathbf{D}$-module, where $\mathbf{D}$ is the divided power algebra over $\mathbf{k}$ in a single variable, and moreover, this $\mathbf{D}$-module is "nearly" finitely presented. This immediately recovers the periodicity result when $\mathbf{k}$ is a field, but also shows, for example, how the torsion varies with $n$ when $\mathbf{k}=\mathbf{Z}$. Using the theory of connections on $\mathbf{D}$-modules, we establish sharp bounds on the period in the case where $\mathbf{k}$ is a field. We apply our theory to obtain results on the modular cohomology of Specht modules and the integral cohomology of unordered configuration spaces of manifolds.