Homology of FI-modules

Abstract

We prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module V in terms of the degrees of its generators and relations. We use this to refine a result of Putman on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics and to integral homology while maintaining explicit bounds for the stable range.

34 pages. v2: major reorganization; to appear in Geometry and Topology

References in corpus (3)

• A long exact sequence for homology of FI-modules
• Upper bounds of homological invariants of $F{I}_G$-modules
• Uniformly Presented Vector Spaces

Cited by in corpus (46)

• FI- and OI-modules with varying coefficients
• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• Higher order representation stability and ordered configuration spaces of manifolds
• Homological Invariants of FI-modules and FI_G-modules
• A long exact sequence for homology of FI-modules
• Regularity of FI-modules and local cohomology
• Periodicity in the cohomology of symmetric groups via divided powers
• Upper bounds of homological invariants of $F{I}_G$-modules
• Homological degrees of representations of categories with shift functors
• Depth and the Local Cohomology of FI_G-modules
• Homological stability of topological moduli spaces
• On the degree-wise coherence of FI_G-modules
• Stability in the high-dimensional cohomology of congruence subgroups
• Filtrations and Homological degrees of FI-modules
• Bounding the homology of FI-modules
• Invariant chains in algebra and discrete geometry
• Central stability homology
• Representation stability, secondary stability, and polynomial functors
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups
• Linear stable range for homology of congruence subgroups via FI-modules
• FI${}^m$-modules over Noetherian rings
• A generating function approach to new representation stability phenomena in orbit configuration spaces
• Two homological proofs of the Noetherianity of $F{I}_G$
• A remark on FI-module homology
• Regularity bounds for twisted commutative algebras
• On stable homology of congruence groups
• GL-algebras in positive characteristic I: the exterior algebra
• FI-hyperhomology and ordered configuration spaces
• Bounds on homological invariants of VI-modules
• Homological finiteness of functors on an additive category and applications
• Classical homological stability from the point of view of cells
• Polynomial conditions and homology of FI-modules
• Regularity and stable ranges of FI-modules
• Generalized Representation Stability and FI_d-modules
• A Survey of Representation Stability Theory
• On computing the eventual behavior of an FI-module over the rational numbers
• Local cohomology and the multi-graded regularity of FI${}^m$-modules
• Sheaves of modules on atomic sites and discrete representations of topological groups
• On the tails of FI-modules
• Castelnuovo-Mumford regularity of representations of certain product categories
• An inductive machinery for representations of categories with shift functors
• An inductive method for $\mathrm{OI}$-modules
• Onset of regularity for FI-modules
• Asymptotic behaviors of representations of graded categories with inductive functors
• Representation stability and arithmetic statistics of spaces of 0-cycles