FI-modules over Noetherian ringsReport as inadecuate

 FI-modules over Noetherian rings

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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 an integral version of Putmans central stability for homology of congruence subgroups.

Author: Thomas Church; Jordan S. Ellenberg; Benson Farb; Rohit Nagpal


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