Algorithms for graded injective resolutions and local cohomology over semigroup rings

Let Q be an affine semigroupgenerating , and fix a finitely generated -graded module M over the semigroup algebra for a field . We provide an algorithm to compute a minimal -graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modules supported on any monomial (that is, -graded) ideal I. Since these local cohomology modules are neither finitely generated nor finitely cogenerated, part of this task is defining a finite data structure to encode them.