Graded local cohomology of modules over semigroup rings

We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. A Cohen-Macaulay criterion immediately follows. We also provide an alternative proof of a result of [18] characterizing Cohen-Macaulay affine semigroup rings. (c) 2023 Elsevier Inc. All rights reserved.