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FI-modules over Noetherian rings. (English) Zbl 1344.20016
Summary: 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.

MSC:
20C30 Representations of finite symmetric groups
13C60 Module categories and commutative rings
55R80 Discriminantal varieties and configuration spaces in algebraic topology
05E10 Combinatorial aspects of representation theory
20J06 Cohomology of groups
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
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References:
[1] A Borel, J P Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436 · Zbl 0274.22011 · doi:10.1007/BF02566134 · eudml:139559
[2] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1994)
[3] F Calegari, The stable homology of congruence subgroups, · Zbl 1336.11045 · arxiv:1311.5190
[4] R Charney, On the problem of homology stability for congruence subgroups, Comm. Algebra 12 (1984) 2081 · Zbl 0542.20023 · doi:10.1080/00927878408823099
[5] T Church, J S Ellenberg, Homological properties of \(\mathrm{FI}\)-modules and stability, in preparation · Zbl 1371.18012
[6] T Church, J S Ellenberg, B Farb, \(\mathrm{FI}\)-modules: A new approach to stability for \(S_n\)-representations, (2012) · arxiv:1204.4533v2
[7] T Church, B Farb, Representation theory and homological stability, Adv. Math. 245 (2013) 250 · Zbl 1300.20051 · doi:10.1016/j.aim.2013.06.016 · arxiv:1008.1368
[8] T Church, A Putman, Generating the Johnson filtration, · Zbl 1364.20025 · arxiv:1311.7150
[9] R Jimenez Rolland, On the cohomology of pure mapping class groups as \(\mathrm{FI}\)-modules, J. Homotopy Relat. Struct. (2013) · arxiv:1207.6828
[10] W Lück, Transformation groups and algebraic \(K\!\)-theory, Lecture Notes in Mathematics 1408, Springer (1989) · Zbl 0679.57022 · doi:10.1007/BFb0083681
[11] A Putman, Stability in the homology of congruence subgroups, · Zbl 1334.20045 · arxiv:1201.4876v4
[12] A Putman, S Sam, Representation stability and finite linear groups, (2014) · Zbl 1408.18003 · arxiv:1408.3694v1
[13] E Riehl, Categorical homotopy theory, New Mathematical Monographs 24, Cambridge Univ. Press (2014) · Zbl 1317.18001 · doi:10.1017/CBO9781107261457
[14] S Sam, A Snowden, \(\mathrm{GL}\)-equivariant modules over polynomial rings in infinitely many variables, · Zbl 1436.13012 · arxiv:1206.2233v2
[15] A Snowden, Syzygies of Segre embeddings and \(\Delta\)-modules, Duke Math. J. 162 (2013) 225 · Zbl 1279.13024 · doi:10.1215/00127094-1962767 · euclid:dmj/1359036935
[16] B Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996) 1057 · Zbl 0857.57025 · doi:10.1016/0040-9383(95)00058-5
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