Divisorial linear algebra of normal semigroup rings

TL;DR

This paper studies the structure of divisorial ideals in normal semigroup rings, showing finiteness properties and analyzing their minimal generators and depth, with generalizations to systems of linear inequalities and diophantine equations.

Contribution

It establishes finiteness results for divisorial ideals with bounded generators and depth, and connects these properties to Hilbert functions and linear inequalities.

Findings

Finiteness of divisorial ideals with bounded minimal generators

Determination of minimal depth of divisorial ideals

Behavior of generators and depth in divisor class groups

Abstract

We investigate the minimal number of generators $\mu$ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound $C$ there exist, up to isomorphism, only finitely divisorial ideals $I$ such that $\mu(I)\le C$. It follows that there exist only finitely many Cohen--Macaulay divisor classes. Moreover we determine the minimal depth of all divisorial ideals and the behaviour of $\mu$ and depth in ``arithmetic progressions'' in the divisor class group. The results are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the non-negative solutions of an inhomogeneous system of linear diophantine equations. We also give a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals: it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions.