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

TL;DR

This paper develops a local cohomology theory for FI^m-modules, establishing parallels with classical multi-graded module theory, and introduces an invariant that behaves similarly to known invariants in algebraic geometry.

Contribution

It introduces a local cohomology framework for FI^m-modules and defines a new invariant analogous to invariants in multi-graded module theory.

Findings

The local cohomology theory for FI^m-modules closely mimics classical theory.

An invariant for FI^m-modules is defined using this theory.

The invariant behaves similarly to Maclagan and Smith's invariant for multi-graded 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.