The Tor algebra of trimmings of Gorenstein ideals
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909099167318016 |
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| author | Ferraro, Luigi Hardesty, Alexis |
| author_facet | Ferraro, Luigi Hardesty, Alexis |
| contents | Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that $I$ is generated by the sub-maximal pfaffians of this matrix. Let $J$ be the ideal obtained by multiplying some of the pfaffian generators of $I$ by $\mathfrak{m}$; we say that $J$ is a trimming of $I$. Building on a recent paper of Vandebogert, we construct an explicit free resolution of $R/J$ and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class $\mathbf{G}$ hold true in our context. Furthermore, we address the realizability question for ideals of class $\mathbf{G}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_05228 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Tor algebra of trimmings of Gorenstein ideals Ferraro, Luigi Hardesty, Alexis Commutative Algebra 13D02, 13D07, 13H10 Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that $I$ is generated by the sub-maximal pfaffians of this matrix. Let $J$ be the ideal obtained by multiplying some of the pfaffian generators of $I$ by $\mathfrak{m}$; we say that $J$ is a trimming of $I$. Building on a recent paper of Vandebogert, we construct an explicit free resolution of $R/J$ and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class $\mathbf{G}$ hold true in our context. Furthermore, we address the realizability question for ideals of class $\mathbf{G}$. |
| title | The Tor algebra of trimmings of Gorenstein ideals |
| topic | Commutative Algebra 13D02, 13D07, 13H10 |
| url | https://arxiv.org/abs/2204.05228 |