A generalization of Maloo's theorem on freeness of derivation modules

Cleto B. Miranda-Neto and Thyago S. Souza

Vol. 302 (2019), No. 2, 693–708

DOI: 10.2140/pjm.2019.302.693

Abstract

Let $A$ be a Noetherian local $k$-domain ($k$ is a Noetherian ring) whose derivation module ${Der}_k\left(A\right)$ 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 $height{\mathfrak{P}}_{A∕k}\le 2$, then ${Der}_k\left(A\right)$ is $A$-free. In this note we prove the following generalization: if ${projdim}_A\left({Der}_k\left(A\right)\right)<\infty$ and $grade{\mathfrak{P}}_{A∕k}=height{\mathfrak{P}}_{A∕k}\le 2$, then ${Der}_k\left(A\right)$ 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 ${Der}_k\left(A\right)$ 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 $depthA\le 2$ in a much more general context.

Keywords

derivation, maximally differential ideal, projective dimension, Herzog–Vasconcelos conjecture

Mathematical Subject Classification 2010

Primary: 13N15, 13D05, 13C10

Secondary: 13H10, 13C13, 13G05

Milestones

Received: 3 October 2018

Accepted: 13 March 2019

Published: 27 November 2019