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) |
Keywords:
sequences of representations of symmetric groups; FI-modules; representation stability; congruence subgroups; configuration spaces; cohomologyReferences:
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