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

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

Geometry & Topology 18 (2014) 2951–2984

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.

FI-modules, representation stability, congruence subgroup, configuration space, cohomology
Mathematical Subject Classification 2010
Primary: 20B30
Secondary: 20C32
Received: 2 April 2013
Revised: 5 March 2014
Accepted: 4 April 2014
Published: 1 December 2014
Proposed: Walter Neumann
Seconded: Ralph Cohen, Jesper Grodal
Thomas Church
Department of Mathematics
Stanford University
450 Serra Mall
Stanford, CA 94305
Jordan S Ellenberg
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706
Benson Farb
Department of Mathematics
University of Chicago
5734 University Avenue
Chicago, IL 60637
Rohit Nagpal
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706