On the degree-wise coherence of FI_G-modules

Abstract

In this work we study a kind of coherence condition on FI_G-modules, which generalizes the usual notion of finite generation. We prove that a module is coherent, in the appropriate sense, if and only if its generators, as well as its torsion, appears in only finitely many degrees. Using this technical result, we prove that the category of coherent FI_G-modules is abelian, independent of any assumptions on the group G, or the coefficient ring k. Following this, we consider applications towards the local cohomology theory of FI_G-modules, introduced by Li and the author in previous work.

References in corpus (13)

• FI-modules over Noetherian rings
• 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
• A long exact sequence for homology of FI-modules
• Homological Invariants of FI-modules and FI_G-modules
• 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
• Filtrations and Homological degrees of FI-modules
• Two homological proofs of the Noetherianity of $F{I}_G$
• A remark on FI-module homology

Cited by in corpus (6)

• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups
• FI${}^m$-modules over Noetherian rings
• Regularity and stable ranges of FI-modules
• Castelnuovo-Mumford regularity of representations of certain product categories
• Asymptotic behaviors of representations of graded categories with inductive functors
• An inductive machinery for representations of categories with shift functors