Sheaves on finite posets and modules over normal semigroup rings

Recently, the local cohomology module HIi(S) of a polynomial ring S with supports in a monomial ideal I has been studied by several authors. In the present paper, we will extend these results to a normal Gorenstein semigroup ring R=k[xccset membership, variantC] of image. More precisely, we will study the local cohomology modules HIi(R) with supports in monomial ideals I, and their injective resolutions. Roughly speaking, we will see that they only depend on the combinatorial properties of the face lattice of a polytope associated to R. Hence, if R is simplicial, it behaves just like a polynomial ring in our context. For example, the Bass numbers of HIi(R) are always finite in the simplicial case. If R is not simplicial, this is not true as a famous example of Hartshorne shows.