A GENERALIZATION OF MALOO'S THEOREM ON FREENESS OF DERIVATION MODULES

摘要

Let A be a Noetherian local k-domain (k is a Noetherian ring) whose derivation module Der_k (A) is finitely generated as an A-module, and let B_(A/k) ? A be the corresponding maximally differential ideal. A theorem due to Maloo states that if A is regular and heightPA= k ≤ 2, then Der_k (A) is A-free. In this note we prove the following generalization: if projdimA.Der_k(A) <1 and grade B_(A/k) = height B_(A/k) ≤ 2, then Derk.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 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 depth A ≤ 2 in a much more general context.