Structure and Symmetry of Sally Type Semigroup Rings

TL;DR

This paper studies Sally type semigroup rings, revealing their ideal structures, Gorenstein conditions, and providing explicit resolutions and Betti number computations for certain cases.

Contribution

It characterizes Sally type semigroups' Gorenstein property, describes their ideal structures, and constructs explicit minimal resolutions.

Findings

Semigroup rings' ideals often decompose into sums of determinantal ideals.

Gorenstein property of Sally type semigroups occurs precisely when deleting k consecutive integers.

Explicit minimal free resolutions and Betti numbers are provided for specific semigroup rings.

Abstract

Consider a numerical semigroup minimally generated by a subset of the interval $[e,2e-1]$ with multiplicity $e$ and width $e-1$. Such numerical semigroups are called Sally type semigroups. We show that the defining ideals of these semigroup rings, when the embedding dimension is $e-2$, generically have the structure of the sum of two determinantal ideals. More generally, Sally type numerical semigroups with multiplicity $e$ and embedding dimension $d=e-k$ are obtained by introducing $k$ gaps in the interval $[e,2e-1]$. It is known that for $k =2$, there is precisely one such semigroup that is Gorenstein, and it happens when one deletes consecutive integers. Let $S^e_k(j)$ denote the Sally type numerical semigroup of multiplcity $e$, embedding dimension $e-k$ obtained by deleting the $k$ consecutive integers $j, j+1, \ldots, j+k-1$.We prove that for any $1\le k < e/2$, the semigroup $S^e_k(j)$ is Gorenstein if and only if $j=k$. We construct an explicit minimal free resolution of the semigroup ring of $S^e_k(k)$ and compute the Betti numbers. In general, we characterize when $S^e_k(j)$ are symmetric and construct minimal resolutions for these Gorenstein semigroup rings.