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

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.