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A generalization of Maloo’s theorem on freeness of derivation modules. (English) Zbl 1439.13072
Summary: Let $$A$$ be a Noetherian local $$k$$-domain $$(k$$ is a Noetherian ring) whose derivation module $$\operatorname{Der}_k(A)$$ is finitely generated as an $$A$$-module, and let $$\mathfrak{P}_{A/k}\subset A$$ be the corresponding maximally differential ideal. A theorem due to Maloo states that if $$A$$ is regular and $$\operatorname{height}\mathfrak{P}_{A/k}\leq 2$$, then $$\operatorname{Der}_k(A)$$ is $$A$$-free. In this note we prove the following generalization: if $$\operatorname{projdim}_A(\operatorname{Der}_k(A))<\infty$$ and $$\operatorname{grade}\mathfrak{P}_{A/k}=\operatorname{height}\mathfrak{P}_{A/k}\leq 2$$, then $$\operatorname{Der}_k(A)$$ is $$A$$-free. We provide several corollaries – to wit, the cases where $$A$$ contains a field of positive characteristic, $$A$$ is Cohen-Macaulay, or $$A$$ is a factorial domain – as well as examples with $$\operatorname{Der}_k(A)$$ having infinite projective dimension. Moreover, our result connects to the Herzog-Vasconcelos conjecture, raised for algebras essentially of finite type over a field of characteristic zero, which we show to be true if $$\operatorname{depth}A\leq 2$$ in a much more general context.
##### MSC:
 13N15 Derivations and commutative rings 13D05 Homological dimension and commutative rings 13C10 Projective and free modules and ideals in commutative rings 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13C13 Other special types of modules and ideals in commutative rings 13G05 Integral domains
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