Vol. 302, No. 2, 2019

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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
Abstract

Let A be a Noetherian local k-domain (k is a Noetherian ring) whose derivation module Derk(A) is finitely generated as an A-module, and let PAk A be the corresponding maximally differential ideal. A theorem due to Maloo states that if A is regular and heightPAk 2, then Derk(A) is A-free. In this note we prove the following generalization: if projdimA(Derk(A)) < and gradePAk = heightPAk 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 Derk(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 depthA 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
Authors
Cleto B. Miranda-Neto
Departamento de Matemática
Universidade Federal da Paraíba
João Pessoa, PB
Brazil
Thyago S. Souza
Departamento de Matemática
Universidade Federal da Paraíba
João Pessoa, PB
Brazil