Ishida complex and graded local cohomology of modules over semigroup rings

In this talk, we will discuss the Ishida complex and use this to calculate local cohomology (supported at the graded maximal ideal) for modules over affine semigroup rings. To classify the graded pieces of the Ishida complex we talk about degree space, degree pair topology, grains, and chaff. This combinatorial framework also provides Hochster-type formulas for the Hilbert series of local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. Our main goal is to discuss all these constructions with plenty of examples.