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$\mathrm{FI}$-modules
over Noetherian rings
Thomas Church, Jordan S Ellenberg, Benson Farb and Rohit Nagpal |
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Geometry & Topology 18 (2014) 2951–2984
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Abstract |
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-modules were introduced by the first three authors to encode sequences of representations of symmetric groups. Over a field of characteristic , finite generation of an -module implies representation stability for the corresponding sequence of –representations. In this paper we prove the Noetherian property for -modules over arbitrary Noetherian rings: any sub--module of a finitely generated -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 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
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Mathematical Subject Classification 2010
Primary: 20B30
Secondary: 20C32
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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
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Authors |
| Thomas Church | |
| Department of Mathematics Stanford University 450 Serra Mall Stanford, CA 94305 USA |
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| http://math.stanford.edu/~church | |
| Jordan S Ellenberg | |
| Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706 USA |
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| http://www.math.wisc.edu/~ellenber/ | |
| Benson Farb | |
| Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637 USA |
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| http://www.math.uchicago.edu/~farb/ | |
| Rohit Nagpal | |
| Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706 USA |
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| http://www.math.wisc.edu/~nagpal/ | |
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