Local cohomology and the multigraded regularity of ℱℐm-modules

We develop a local cohomology theory for ℱℐm-modules, and show that it in many ways mimics the classical theory for multigraded modules over a polynomial ring. In particular, we define an invariant of ℱℐm-modules using this local cohomology theory which closely resembles an invariant of multigraded 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.