Bass numbers of semigroup-graded local cohomology

Abstract

Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology H i I(M) at any graded ideal I in terms of Ext modules. We use this method to obtain finiteness results for the local cohomology of graded modules over semigroup rings. In particular we prove that for a semigroup Q whose saturation Q sat is simplicial, and a finitely generated module M over k[Q] that is graded by Q gP , the Bass numbers of H i I(M) are finite for any Q-graded ideal I of k[Q]. Conversely, if Q sat is not simplicial, we find a graded ideal I and graded k[Q]-module M such that the local cohomology module H i I(M) has infinite-dimensional socle. We introduce and exploit the combinatorially defined essential set of a semigroup.