Volume 18, issue 5 (2014)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
$\mathrm{FI}$-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 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.

Keywords
FI-modules, representation stability, congruence subgroup, configuration space, cohomology
Mathematical Subject Classification 2010
Primary: 20B30
Secondary: 20C32
References
Publication
Received: 2 April 2013
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/