Cohomology on Toric Varieties and Local Cohomology with Monomial Supports

In this note we describe aspects of the cohomology of coherent sheaves on a complete toric variety X over a field k and, more generally, the local cohomology, with supports in a monomial ideal, of a finitely generated module over a polynomial ring S. This leads to an efficient way of computing such cohomology, for which we give explicit algorithms. The problem is finiteness. The ith local cohomology of an S-module P with supports in an ideal B is the limit HB(P ) = lim −→ ` Ext(S/B`, P ),