FI-modules over Noetherian rings

Church, Thomas¹ ; Ellenberg, Jordan S² ; Farb, Benson³ ; Nagpal, Rohit²

¹ Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, CA 94305, USA
² Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, USA
³ Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637, USA

Geometry & topology, Tome 18 (2014) no. 5, pp. 2951-2984.

Résumé

$FI$-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.

Détail

DOI : 10.2140/gt.2014.18.2951

Classification : 20B30, 20C32
Keywords: FI-modules, representation stability, congruence subgroup, configuration space, cohomology

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