Combinatorial multigraded book-keeping and the Ishida complex

Abstract: Local cohomology is one of the several homological tools to determine whether a module is Cohen-Macaulay. In the case of modules over semigroup rings, local cohomology (at the maximal ideal) can be computed using the Ishida complex. It is standard to look at each multigraded component of the Ishida complex individually, as the corresponding cohomology reduces to the cohomology of a polyhedral complex. These polyhedral complexes depend on the multidegree, but in fact, many multidegrees give rise to the same polyhedral complex. Our main contribution is a combinatorial way to keep track of these coincidences. As a consequence, we obtain a Cohen-Macaulayness criterion for monomial ideals in semigroup rings that involves cohomology of finitely topological spaces, and an alternative proof of a result of Trung and Hoa classifying Cohen-Macaulay semigroup rings.

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