Algorithms for graded injective resolutions and local cohomology over semigroup rings

Let Q be an affine semigroup generating Z, and fix a finitely generated Z -graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modulesH I(M) supported on any monomial (that is, Z -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.