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On one-dimensional local rings and Berger’s conjecture
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 2 / May 2023
- Published online by Cambridge University Press:
- 23 May 2023, pp. 437-452
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- Article
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Let k be a field of characteristic zero and let
$\Omega_{A/k}$ be the universally finite differential module of a k-algebra A, which is the local ring of a closed point of an algebraic or algebroid curve over k. A notorious open problem, known as Berger’s Conjecture, predicts that A must be regular if 
$\Omega_{A/k}$ is torsion-free. In this paper, assuming the hypotheses of the conjecture and observing that the module 
${\rm Hom}_A(\Omega_{A/k}, \Omega_{A/k})$ is then isomorphic to an ideal of A, say 
$\mathfrak{h}$, we show that A is regular whenever the ring 
$A/a\mathfrak{h}$ is Gorenstein for some parameter a (and conversely). In addition, we provide various characterizations for the regularity of A in the context of the conjecture.
