FI-modules over Noetherian rings

Abstract

FI-modules were introduced by the first three authors in [CEF] 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 of the results of [CEF] 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.

32 pages; v2: reorganized paper, expanded introduction and added Theorems B and C

References in corpus (4)

• FI-modules and stability for representations of symmetric groups
• GL-equivariant modules over polynomial rings in infinitely many variables
• Representation stability and finite linear groups
• On the cohomology of pure mapping class groups as FI-modules

Cited by in corpus (72)

• FI-modules and stability for representations of symmetric groups
• Gröbner methods for representations of combinatorial categories
• Representation stability and finite linear groups
• Homology of FI-modules
• FI-modules and the cohomology of modular representations of symmetric groups
• Representations of categories of G-maps
• Noetherianity of some degree two twisted commutative algebras
• A spectral sequence for stratified spaces and configuration spaces of points
• FI- and OI-modules with varying coefficients
• Noetherian property of infinite EI categories
• Noetherianity of some degree two twisted skew-commutative algebras
• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• Homological Invariants of FI-modules and FI_G-modules
• A long exact sequence for homology of FI-modules
• Higher order representation stability and ordered configuration spaces of manifolds
• Periodicity in the cohomology of symmetric groups via divided powers
• Regularity of FI-modules and local cohomology
• Upper bounds of homological invariants of $F{I}_G$-modules
• Representation stability for cohomology of configuration spaces in ${\mathbf{R}}^d$
• Homological degrees of representations of categories with shift functors
• Depth and the Local Cohomology of FI_G-modules
• The representation theory of Brauer categories I: triangular categories
• Homological stability of topological moduli spaces
• On the degree-wise coherence of FI_G-modules
• Stability in the high-dimensional cohomology of congruence subgroups
• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• Stability and periodicity in the modular representation theory of symmetric groups
• FI_W-modules and stability criteria for representations of the classical Weyl groups
• Filtrations and Homological degrees of FI-modules
• Generating the Johnson filtration
• Combinatorial categorical equivalences of Dold-Kan type
• Algebraic structures on cohomology of configuration spaces of manifolds with flows
• Koszulity of directed categories in representation stability theory
• Uniformly Presented Vector Spaces
• Coinduction functor in representation stability theory
• On the cohomology of pure mapping class groups as FI-modules
• FI_W-modules and constraints on classical Weyl group characters
• Central stability homology
• Invariant chains in algebra and discrete geometry
• Stability results for Houghton groups
• An application of the theory of FI-algebras to graph configuration spaces
• Stability in the homology of unipotent groups
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups
• FI${}^m$-modules over Noetherian rings
• A remark on FI-module homology
• Two homological proofs of the Noetherianity of $F{I}_G$
• Character Polynomials and the Restriction Problem
• Linear stable range for homology of congruence subgroups via FI-modules
• On stable homology of congruence groups
• Unital operads, monoids, monads, and bar constructions
• The artinian conjecture (following Djament, Putman, Sam, and Snowden)
• Commuting matrices and Atiyah's Real K-theory
• Representation stability and outer automorphism groups
• FI-hyperhomology and ordered configuration spaces
• Representation stability for sequences of 0-Hecke modules
• On finite generation in magnitude (co)homology, and its torsion
• VIC-modules over noncommutative rings
• Generalized Representation Stability and FI_d-modules
• A Survey of Representation Stability Theory
• Regularity and stable ranges of FI-modules
• Polynomial conditions and homology of FI-modules
• Adjoint functors on the representation category of $\mathcal{OI}$
• Local cohomology and the multi-graded regularity of FI${}^m$-modules
• On the tails of FI-modules
• Sheaves of modules on atomic sites and discrete representations of topological groups
• Representation stability for homotopy automorphisms
• An inductive machinery for representations of categories with shift functors
• Homological stability for quotients of mapping class groups of surfaces by the Johnson subgroups
• Asymptotic behaviors of representations of graded categories with inductive functors
• Castelnuovo-Mumford regularity of representations of certain product categories
• An inductive method for $\mathrm{OI}$-modules