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
), where B is any sequence of ideals that is cofinal with the powers of B. We will be interested in the case where S is a polynomial ring, P is a finitely generated module, and B is a monomial ideal. The module on the left of this equality is almost never finitely generated (even when P = S), whereas the module Ext i (S/B , P ) on the right is finitely generated, so that the limit is really necessary.
We can sometimes restore finiteness by considering the homogeneous components of H i B (P ) for a suitable grading. For example, in the case where B is a monomial ideal and P = S the modules on both sides are Z n graded, and for each p ∈ Z n the component H i B (S) p is a finite-dimensional vector space. From this and the Hilbert syzygy theorem, one easily sees that the corresponding finiteness holds for any finitely generated Z ngraded module P .
To obtain a computation that works for more than Z n -graded modules, we work with coarser gradings; that is we consider a grading in an abelian group D that is a homomorphic image of Z n . Of course if the grading is too coarse, we will lose finiteness again. In this paper we give a criterion on the grading for such finiteness to hold. When it holds we study the convergence of the limit above, and show how to compute, for each δ ∈ D, an explicit ideal B [ ] such that the natural map Ext i (S/B [ ] , P ) δ -→ H i B (P ) δ is an isomorphism. The computation involves the solution of a linear programming problem that depends on the grading D and on the syzygies of P .
The results in this paper were motivated by the problem of computing the cohomology of coherent sheaves on a complete toric variety, and such varieties provide the most