Sheaves on finite posets and modules over normal semigroup rings

Abstract

Recently, the local cohomology module H I i ( 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[ x c | c∈ C] of C⊂ Z d . More precisely, we will study the local cohomology modules H I i ( 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 H I i ( R) are always finite in the simplicial case. If R is not simplicial, this is not true as a famous example of Hartshorne shows.