Volume 18, issue 5 (2014)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 27
Issue 9, 3387–3831
Issue 8, 2937–3385
Issue 7, 2497–2936
Issue 6, 2049–2496
Issue 5, 1657–2048
Issue 4, 1273–1655
Issue 3, 823–1272
Issue 2, 417–821
Issue 1, 1–415

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
 
Other MSP Journals
$\mathrm{FI}$-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/