Let R be a Noetherian local k-algebra whose derivation module \({\mathrm{Der}}_k(R)\) is finitely generated. Our main goal in this paper is to investigate the impact of assuming that \({\mathrm{Der}}_k(R)\) has finite projective dimension (or finite Gorenstein dimension), mainly in connection with freeness, under a suitable hypothesis concerning the vanishing of (co)homology or the depth of a certain tensor product. We then apply some of our results towards the critical case \({\mathrm{depth}}\,R=3\) of the Herzog–Vasconcelos conjecture and consequently to the strong version of the Zariski–Lipman conjecture.

令R为Noetherian局部k-代数，其衍生模块\（{\ mathrm {Der}} _ k（R）\）是有限生成的。本文的主要目标是在适当的假设下研究假设\（{\ mathrm {Der}} _ k（R）\）具有有限的投影维数（或有限的Gorenstein维数），主要与自由有关的影响。关于（共）同构关系的消失或某个张量积的深度。然后，我们将某些结果应用于Herzog-Vasconcelos猜想的临界情况\（{\ mathrm {depth}} \，R = 3 \），并由此应用Zariski-Lipman猜想的强版本。