Computing Injective Resolutions and Local Cohomology

Although injective resolutions are fundamental objects in homological algebra, their computation remains intractable over general Noetherian rings. In the case of affine semigroup rings, Helm and Miller introduced algorithms to compute injective resolutions of finitely generated ${\mathbb{Z}}^d$-graded modules and, as an application, algorithms to compute local cohomology modules. These algorithms exploit the polyhedral structure of ${\mathbb{Z}}^d$-graded injective modules over these rings. We provide an OSCAR implementation of these algorithms, enabling the computation of first injective resolutions and local cohomology modules over non-regular rings.