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

TL;DR

This paper constructs a minimal free resolution of the module of derivations on a determinantal ring using differential graded algebra techniques, providing explicit generators and new insights into its structure.

Contribution

It introduces a novel method employing the relative bar resolution and differential graded structures to explicitly resolve the derivation module on a determinantal ring.

Findings

Explicit minimal generating set for Der_{R|k} obtained

Constructed minimal graded free resolution using differential graded algebra

Provided detailed action of Hilbert-Burch algebra on resolving modules

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.