FI-modules over Noetherian rings

By Thomas Church, Jordan S. Ellenberg, Benson Farb, Rohit Nagpal. 2012

189 citations recorded by the source corpus.

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. 20B30; 20C32

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9eaf7f2915ec5b64cc6a4656cf27e6ea519be450

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Citations in the corpus

FI_W-modules and stability criteria for representations of the classical Weyl groups (2013, methodology): “Shortly after the appearance of Putman’s work, the work on FI–modules [CEF12] were strengthened further by Church–Ellenberg– Farb–Nagpal [CEFN14].”
FI-modules and stability for representations of symmetric groups (2012, background): “an [Pu] proposes an alternative representation stability condition called \central stability" and proves that it holds for the mod-pcohomology of congruence groups. We prove with Rohit Nagpal in [CEFN] that in the language of FI-modules, central stability is equivalent to nite presentation. We ”
FI- and OI-modules with varying coefficients (2017, background): “(ii) In the case of constant coefficients, where A = X is the “constant” FI-algebra over K, that is, An = K for each n, the category FI-Mod(A) is exactly the category of FI-modules over K as, for example, studied in [2, 3, 4].”
FI-modules and the cohomology of modular representations of symmetric groups (2015, methodology): “Πto Ua. The K-modules (kerϕa)m i ⊂ K[Sm i ] 0 U˜a S +a V M˜ (m i) 0 0 Ua S +aV A a 0 0 0 0 Πa S+a ϕa Figure 2 are increasing in aand hence must stabilizes for alarge enough (see Proof of Theorem A in [CEFN] for a proof that these modules are increasing). Fixing such an a, kerϕais generated in degree”
Representation stability and finite linear groups (2014, methodology): “ representation theory of the symmetric group. The key technical innovation of their work was a certain Noetherian property of FI-modules, which they later established in great generality with Nagpal [ChEFaNag]. This Noetherian property allowed them to prove an asymptotic structure theorem for ﬁnite”
Homology of FI-modules (2015, methodology): “M / I Z / , but its formulation here is a combination of [12, Theorem B] and our earlier theorem with Farb and Nagpal [5, Theorem 1.6].”
Higher-order representation stability and ordered configuration spaces of manifolds (2016, unknown): “The main theorem of Church–Ellenberg–Farb–Nagpal [10] shows that the answer is yes for d D 1 .”