Homological Invariants of FI-modules and FI_G-modules

Abstract

We explore a theory of depth for FI_G-modules which are presented in finite degrees. Using this theory, we prove results about the regularity, and provide novel bounds on stable ranges of FI-modules, making effective a theorem of Nagpal and thereby refining the stable range in results of Church, Ellenberg, and Farb.

29 pages. v3: Generalized many of the theorems from the earlier versions. No longer require that G be finite, nor that k be a Noetherian ring, for instance

References in corpus (2)

• FI-modules and the cohomology of modular representations of symmetric groups
• Filtrations and Homological degrees of FI-modules

Cited by in corpus (14)

• Upper bounds of homological invariants of $F{I}_G$-modules
• Depth and the Local Cohomology of FI_G-modules
• On the degree-wise coherence of FI_G-modules
• Filtrations and Homological degrees of FI-modules
• Bounding the homology of FI-modules
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups
• Two homological proofs of the Noetherianity of $F{I}_G$
• Generalized Representation Stability and FI_d-modules
• On computing the eventual behavior of an FI-module over the rational numbers
• Local cohomology and the multi-graded regularity of FI${}^m$-modules
• An inductive machinery for representations of categories with shift functors
• Representation stability and arithmetic statistics of spaces of 0-cycles
• An inductive method for $\mathrm{OI}$-modules
• The commuting complex of the symmetric group with bounded number of $p$-cycles