Structure and Symmetry of Sally Type Semigroup Rings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917226710302720 |
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| author | Singh, Srishti Srinivasan, Hema |
| author_facet | Singh, Srishti Srinivasan, Hema |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_18136 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure and Symmetry of Sally Type Semigroup Rings Singh, Srishti Srinivasan, Hema Commutative Algebra Rings and Algebras Primary 13D02, 13D05, Secondary 20M14, 13H10 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. |
| title | Structure and Symmetry of Sally Type Semigroup Rings |
| topic | Commutative Algebra Rings and Algebras Primary 13D02, 13D05, Secondary 20M14, 13H10 |
| url | https://arxiv.org/abs/2512.18136 |