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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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