Structure and Symmetry of Sally Type Semigroup Rings

TL;DR

The paper addresses the problem of understanding Sally type numerical semigroups generated from interval subsets by removing gaps, focusing on when their semigroup rings are symmetric (Gorenstein) and how to describe their defining ideals and resolutions. It develops a determinantal framework: for embedding dimension $e-2$ the defining ideal is the sum of two 2x2 minor ideals, with Gastinger's criterion ensuring equality to a binomial determinantal ideal; symmetry is completely characterized for $k\le e/2$ (generally $j=k$, with a notable exception when $j=1$ and $e=2k$) and analyzed for $k>e/2$, where symmetry can occur at specific even levels under a level-compatibility condition $n(k-j)=(n-1)(e-2)$. Minimal free resolutions are constructed via the mapping cylinder of the dual Eagon–Northcott resolution, yielding explicit Betti numbers in the symmetric cases, and the approach extends to the $j=1$ case with analogous Betti patterns. The results provide a coherent picture linking determinantal presentations, symmetry criteria, and homological invariants (Frobenius numbers and genus) for Sally type semigroup rings, with potential implications for Cohen–Macaulay and Gorenstein properties in low-dimensional monomial curves.

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.

Figures (3)

• Figure 1: Frobenius Number and Genus for $S_{k}(j)$ given \ref{['eq:2k>e']}.
• Figure 2: Gaps and Nongaps in $S_{k}(j)$
• Figure :

Theorems & Definitions (36)

• Definition 2.1
• Remark 2.2
• Theorem 2.3: Gastinger
• proof
• Theorem 2.4
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof