Graded local cohomology of modules over semigroup rings

TL;DR

This paper provides a combinatorial approach to understanding local cohomology modules over semigroup rings, offering formulas for their Hilbert series and criteria for Cohen--Macaulayness, with implications for algebraic and geometric properties.

Contribution

It introduces a new combinatorial framework for local cohomology of modules over semigroup rings, including Hochster-type formulas and Cohen--Macaulay criteria.

Findings

Hochster-type formulas for Hilbert series of local cohomology modules

A combinatorial criterion for Cohen--Macaulayness of semigroup rings

An alternative proof of Cohen--Macaulay characterization for affine semigroup rings

Abstract

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.