Abstract
-modules were introduced by the first three authors to encode sequences of representations of symmetric groups. Over a field of characteristic , finite generation of an -module implies representation stability for the corresponding sequence of –representations. In this paper we prove the Noetherian property for -modules over arbitrary Noetherian rings: any sub--module of a finitely generated -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 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.
Citation
Thomas Church. Jordan S Ellenberg. Benson Farb. Rohit Nagpal. "$\mathrm{FI}$-modules over Noetherian rings." Geom. Topol. 18 (5) 2951 - 2984, 2014. https://doi.org/10.2140/gt.2014.18.2951
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