FI-modules over Noetherian rings

TL;DR

This paper proves that FI-modules over Noetherian rings are Noetherian, enabling the extension of representation stability results to positive characteristic and integral coefficients, with applications in topology and algebra.

Contribution

It establishes the Noetherian property for FI-modules over arbitrary Noetherian rings, broadening the scope of representation stability theory.

Findings

Proved Noetherian property for FI-modules over Noetherian rings.

Extended representation stability results to positive characteristic and integral coefficients.

Applied the main theorem to cohomology of configuration spaces, coinvariant algebras, and homology of congruence subgroups.

Abstract

FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely generated FI-module is finitely generated. This lets us extend many of the results of [CEF] to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman's central stability for homology of congruence subgroups.