$\mathrm{FI}$-modules over Noetherian rings

Abstract

$FI$-modules were introduced by the first three authors 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 results 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.