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 HIi(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 Qsat is simplicial, and
a finitely generated module M over k[Q] that is graded by Qgp, the Bass
numbers of HIi(M) are finite for any Q-graded ideal I of k[Q]. Conversely, if
Qsat is not simplicial, we find a graded ideal I and graded k[Q]-module M
such that the local cohomology module HIi(M) has infinite-dimensional
socle. We introduce and exploit the combinatorially defined essential set of a
semigroup.
