Algorithms for graded injactive resolutions and local cohomology over semigroup rings

摘要

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