Local cohomology and the multi-graded regularity of FI${}^m$-modules

Abstract

We develop a local cohomology theory for FI${}^m$-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI${}^m$-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.

References in corpus (3)

• FI-modules and the cohomology of modular representations of symmetric groups
• Bounding the homology of FI-modules
• FI${}^m$-modules over Noetherian rings