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FI-modules over Noetherian rings. (English) Zbl 1344.20016
Summary: 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.

##### MSC:
 20C30 Representations of finite symmetric groups 13C60 Module categories and commutative rings 55R80 Discriminantal varieties and configuration spaces in algebraic topology 05E10 Combinatorial aspects of representation theory 20J06 Cohomology of groups 20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
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