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Torsion in differentials and Berger’s conjecture

Abstract

Let \((R,{\mathfrak {m}},\mathbb {k})\) be an equicharacteristic one-dimensional complete local domain over an algebraically closed field \(\mathbb {k}\) of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials \(\Omega _R\) is a torsion-free R module. We give new cases of this conjecture by extending works of Güttes (Arch Math 54:499–510, 1990) and Cortiñas et al. (Math Z 228:569–588, 1998). This is obtained by constructing a new subring S of \({\text {Hom}}_R({\mathfrak {m}},{\mathfrak {m}})\) and constructing enough torsion in \(\Omega _S\), enabling us to pull back a nontrivial torsion to \(\Omega _R\).

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Correspondence to Vivek Mukundan.

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This paper is dedicated to Jürgen Herzog, whose fundamental research in commutative algebra has inspired researchers for 50 years.

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Huneke, C., Maitra, S. & Mukundan, V. Torsion in differentials and Berger’s conjecture. Res Math Sci 8, 60 (2021). https://doi.org/10.1007/s40687-021-00295-y

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Keywords

  • Module of differentials
  • Berger Conjecture
  • Reduced curves

Mathematics Subject Classification

  • Primary: 13N05
  • Secondary: 13H10