Depth and the Local Cohomology of FI_G-modules

Abstract

In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI_G-modules in terms of torsion degrees of their local cohomology groups.

References in corpus (4)

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

Cited by in corpus (8)

• On the degree-wise coherence of FI_G-modules
• FI${}^m$-modules over Noetherian rings
• Two homological proofs of the Noetherianity of $F{I}_G$
• Local cohomology and the multi-graded regularity of FI${}^m$-modules
• Asymptotic behaviors of representations of graded categories with inductive functors
• An inductive machinery for representations of categories with shift functors
• An inductive method for $\mathrm{OI}$-modules
• Sheaves of modules on atomic sites and discrete representations of topological groups