A long exact sequence for homology of FI-modules

Abstract

We construct a long exact sequence involving the homology of an FI-module. Using the long exact sequence, we give two methods to bound the Castelnuovo-Mumford regularity of an FI-module which is generated and related in finite degree. We also prove that for an FI-module which is generated and related in finite degree, if it has a nonzero higher homology, then its homological degrees are strictly increasing (starting from the first homological degree).

Minor corrections

References in corpus (3)

• FI-modules and the cohomology of modular representations of symmetric groups
• Upper bounds of homological invariants of $F{I}_G$-modules
• Filtrations and Homological degrees of FI-modules

Cited by in corpus (8)

• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• On the degree-wise coherence of FI_G-modules
• Two homological proofs of the Noetherianity of $F{I}_G$
• Linear stable range for homology of congruence subgroups via FI-modules
• FI${}^m$-modules over Noetherian rings
• Bounds on homological invariants of VI-modules
• An inductive machinery for representations of categories with shift functors
• An inductive method for $\mathrm{OI}$-modules