A Generalization of the Ishida Complex with applications

TL;DR

This paper introduces a generalized Ishida complex to compute local cohomology of modules over quotients of polynomial rings by cellular binomial ideals, providing new criteria for Cohen--Macaulayness and relating local cohomology in affine semigroup rings.

Contribution

It extends the Ishida complex framework to cellular binomial ideals and establishes a combinatorial Cohen--Macaulayness criterion for such quotients.

Findings

Provides a combinatorial Cohen--Macaulayness criterion for lattice ideals.

Relates local cohomology of affine semigroup rings to quotients by radical monomial ideals.

Applies to semigroups with cones over a simplex.

Abstract

We construct a generalize Ishida complex to compute the local cohomology with monomial support of modules over quotients of polynomial rings by cellular binomial ideals. As a consequence, we obtain a combinatorial criterion to determine when such a quotient is Cohen--Macaulay. In particular, this gives a Cohen--Macaulayness criterion for lattice ideals. We also prove a result relating the local cohomology with radical monomial ideal support of an affine semigroup ring to the local cohomology with maximal ideal support of the quotient of the affine semigroup ring by the radical monomial ideal. This requires a combinatorial assumption on the semigroup, which holds for (not necessarily normal) semigroups whose cone is the cone over a simplex.