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arXiv · 2406.04223

Resolving the Module of Derivations on an $n \times (n+1)$ Determinantal Ring

Henry Potts-Rubin
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Abstract

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.

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Henry Potts-Rubin. 2025-05-12. Resolving the Module of Derivations on an $n \times (n+1)$ Determinantal Ring. https://doi.org/10.1016/j.jalgebra.2025.04.022

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