Type and Conductor of Simplicial Affine Semigroups

TL;DR

This paper generalizes the concept of pseudo-Frobenius numbers to simplicial affine semigroups, characterizes their Cohen-Macaulay type, and explores bounds and conductor generation related to their algebraic properties.

Contribution

It introduces a new definition of type for simplicial affine semigroups, linking it to Cohen-Macaulay properties and providing bounds based on embedding dimension.

Findings

Type is bounded by 2 for Cohen-Macaulay rings with embedding dimension at most d+2.

Type can be arbitrarily large for higher embedding dimensions.

Provides a generating set for the conductor as an ideal of the normalization.

Abstract

We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring $\mathbb{K}[S]$. We define the type of $S$, $\operatorname{type}$, in terms of some Ap\'ery sets of $S$ and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when $\mathbb{K}[S]$ is Cohen-Macaulay. If $\mathbb{K}[S]$ is a $d$-dimensional Cohen-Macaulay ring of embedding dimension at most $d+2$, then $\operatorname{type}\leq 2$. Otherwise, $\operatorname{type}$ might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of $S$ as an ideal of its normalization.