Geometry & Topology

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

Thomas Church, Jordan S Ellenberg, Benson Farb, and Rohit Nagpal

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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 Sn–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.

Article information

Geom. Topol., Volume 18, Number 5 (2014), 2951-2984.

Received: 2 April 2013
Revised: 5 March 2014
Accepted: 4 April 2014
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20B30: Symmetric groups
Secondary: 20C32: Representations of infinite symmetric groups

FI-modules representation stability congruence subgroup configuration space cohomology


Church, Thomas; Ellenberg, Jordan S; Farb, Benson; Nagpal, Rohit. $\mathrm{FI}$-modules over Noetherian rings. Geom. Topol. 18 (2014), no. 5, 2951--2984. doi:10.2140/gt.2014.18.2951.

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