Homological aspects of derivation modules and critical case of the Herzog–Vasconcelos conjecture

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.