Periodicity in the cohomology of symmetric groups via divided powers

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 ${⨁}_{n\ge 0}{\mathrm{H}}^t(S_n,M_n)$ 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.

Fixed some minor mistakes and expanded the section on configuration spaces

References in corpus (2)

• FI-modules and the cohomology of modular representations of symmetric groups
• Higher order representation stability and ordered configuration spaces of manifolds

Cited by in corpus (8)

• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups
• Divided power algebras over an operad
• A periodicity theorem for extensions of Weyl modules
• On stability and nonvanishing of homomorphism spaces between Weyl modules
• Periodicity in the cohomology of finite general linear groups via q-divided powers
• An inductive method for $\mathrm{OI}$-modules
• Configuration space in a product