Volume 18, issue 5 (2014)

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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
Abstract

$FI$-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.

Keywords
FI-modules, representation stability, congruence subgroup, configuration space, cohomology
Primary: 20B30
Secondary: 20C32
Publication
Revised: 5 March 2014
Accepted: 4 April 2014
Published: 1 December 2014
Proposed: Walter Neumann
Seconded: Ralph Cohen, Jesper Grodal
Authors
 Thomas Church Department of Mathematics Stanford University 450 Serra Mall Stanford, CA 94305 USA http://math.stanford.edu/~church Jordan S Ellenberg Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706 USA http://www.math.wisc.edu/~ellenber/ Benson Farb Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637 USA http://www.math.uchicago.edu/~farb/ Rohit Nagpal Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706 USA http://www.math.wisc.edu/~nagpal/