Local Cohomology, Multigradings and Polyhedral Combinatorics

Abstract

The goal of combinatorial commutative algebra is to study the interplay between commutative algebra and various subfields of combinatorics such as enumerative combinatorics and discrete geometry. Among the central objects in combinatorial commutative algebra are square-free monomial ideals and semigroup rings. This dissertation examines monomial ideals in affine semigroup rings.

The two main results in this dissertation are as follows. First, we give a combinatorial characterization for not only the Cohen–Macaulay Zd-graded module of affine semigroup rings but also the quotients of polynomial rings by cellular binomial ideals. This criterion involves vanishing homology of finitely many polyhedral cell complexes. These polyhedral cell complexes are derived from degree spaces, a space of all multigradings with special finite topology. Our main contribution is to construct degree spaces corresponding to Zd-graded modules.

Next, we elucidate a hidden duality between the local cohomologies of simplicial affine semi-group rings by extending the Ishida complex as well as using a Hochster-type Hilbert series formula. The extensions of the Ishida complex allow us to calculate local cohomology of the given module with all possible radical monomial ideal supports. With our degree spaces, we calculate the Hilbert series of both the local cohomology of the simplicial affine semigroup ring k[Q] with a monomial ideal I support and that of the quotient k[Q]/I with the maximal monomial ideal support. Finally, we showed that there is a 1-1 correspondence of grains between these two local cohomologies such that whose cohomologies of the chaffs are dual.