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Abstract
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Let
be a Noetherian
local
-domain
(
is a Noetherian ring) whose derivation module
is finitely generated
as an
-module,
and let
be the corresponding maximally differential ideal. A theorem due to Maloo states that if
is regular
and
,
then
is
-free.
In this note we prove the following generalization: if
and
, then
is
-free.
We provide several corollaries — to wit, the cases where
contains a field of
positive characteristic,
is Cohen–Macaulay, or
is a factorial domain — as well as examples with
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
in a
much more general context.
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Keywords
derivation, maximally differential ideal, projective
dimension, Herzog–Vasconcelos conjecture
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Mathematical Subject Classification 2010
Primary: 13N15, 13D05, 13C10
Secondary: 13H10, 13C13, 13G05
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Milestones
Received: 3 October 2018
Accepted: 13 March 2019
Published: 27 November 2019
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