Resolving the Module of Derivations on an $n \times (n+1)$ Determinantal Ring
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910939545075712 |
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| author | Potts-Rubin, Henry |
| author_facet | Potts-Rubin, Henry |
| contents | We use the construction of the relative bar resolution via differential graded structures to obtain the minimal graded free resolution of $\text{Der}_{R \mid k}$, where $R$ is a determinantal ring defined by the maximal minors of an $n \times (n+1)$ generic matrix and $k$ is its coefficient field. Along the way, we compute an explicit action of the Hilbert-Burch differential graded algebra on a differential graded module resolving the cokernel of the Jacobian matrix whose kernel is $\text{Der}_{R \mid k}$. As a consequence of the minimality of the resulting relative bar resolution, we get a minimal generating set for $\text{Der}_{R \mid k}$ as an $R$-module, which, while already known, has not been obtained via our methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04223 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Resolving the Module of Derivations on an $n \times (n+1)$ Determinantal Ring Potts-Rubin, Henry Commutative Algebra We use the construction of the relative bar resolution via differential graded structures to obtain the minimal graded free resolution of $\text{Der}_{R \mid k}$, where $R$ is a determinantal ring defined by the maximal minors of an $n \times (n+1)$ generic matrix and $k$ is its coefficient field. Along the way, we compute an explicit action of the Hilbert-Burch differential graded algebra on a differential graded module resolving the cokernel of the Jacobian matrix whose kernel is $\text{Der}_{R \mid k}$. As a consequence of the minimality of the resulting relative bar resolution, we get a minimal generating set for $\text{Der}_{R \mid k}$ as an $R$-module, which, while already known, has not been obtained via our methods. |
| title | Resolving the Module of Derivations on an $n \times (n+1)$ Determinantal Ring |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2406.04223 |