A study of a quadratic almost complete intersection ideal and its linked Gorenstein ideal
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| Format: | Preprint |
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2025
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| author | Diethorn, Rachel Güntürkün, Sema Hardesty, Alexis Mete, Pinar Şega, Liana Sobieska, Aleksandra Veliche, Oana |
| author_facet | Diethorn, Rachel Güntürkün, Sema Hardesty, Alexis Mete, Pinar Şega, Liana Sobieska, Aleksandra Veliche, Oana |
| contents | We examine the ideal $I=(x_1^2, \dots, x_n^2, (x_1+\dots+x_n)^2)$ in the polynomial ring $Q=k[x_1, \dots, x_n]$, where $k$ is a field of characteristic zero or greater than $n$. We also study the Gorenstein ideal $G$ linked to $I$ via the complete intersection ideal $(x_1^2, \dots, x_n^2)$. We compute the Betti numbers of $I$ and $G$ over $Q$ when $n$ is odd and extend known computations when $n$ is even. A consequence is that the socle of $Q/I$ is generated in a single degree (thus $Q/I$ is level) and its dimension is a Catalan number. We also describe the generators and the initial ideal with respect to reverse lexicographic order for the Gorenstein ideal $G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_03977 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A study of a quadratic almost complete intersection ideal and its linked Gorenstein ideal Diethorn, Rachel Güntürkün, Sema Hardesty, Alexis Mete, Pinar Şega, Liana Sobieska, Aleksandra Veliche, Oana Commutative Algebra 13D02, 13C13, 13P10 We examine the ideal $I=(x_1^2, \dots, x_n^2, (x_1+\dots+x_n)^2)$ in the polynomial ring $Q=k[x_1, \dots, x_n]$, where $k$ is a field of characteristic zero or greater than $n$. We also study the Gorenstein ideal $G$ linked to $I$ via the complete intersection ideal $(x_1^2, \dots, x_n^2)$. We compute the Betti numbers of $I$ and $G$ over $Q$ when $n$ is odd and extend known computations when $n$ is even. A consequence is that the socle of $Q/I$ is generated in a single degree (thus $Q/I$ is level) and its dimension is a Catalan number. We also describe the generators and the initial ideal with respect to reverse lexicographic order for the Gorenstein ideal $G$. |
| title | A study of a quadratic almost complete intersection ideal and its linked Gorenstein ideal |
| topic | Commutative Algebra 13D02, 13C13, 13P10 |
| url | https://arxiv.org/abs/2504.03977 |