Resolving the module of derivations on an n × (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 ${\mathrm{Der}}_{R|k}$, 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 ${\mathrm{Der}}_{R|k}$. As a consequence of the minimality of the resulting relative bar resolution, we get a minimal generating set for ${\mathrm{Der}}_{R|k}$ as an R-module, which, while already known, has not been obtained via our methods.