Torsion in Differentials and Berger's Conjecture

TL;DR

This paper advances Berger's conjecture by identifying new cases where the module of differentials has torsion, linking torsion properties to the regularity of one-dimensional local domains over algebraically closed fields.

Contribution

It introduces a novel subring construction to produce torsion in the module of differentials, extending previous results on Berger's conjecture in characteristic zero.

Findings

Established new cases supporting Berger's conjecture

Constructed a subring to generate torsion in differentials

Linked torsion presence to non-regularity of the ring

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 G\"uttes (Arch Math 54:499-510, 1990) and Corti\~nas et al. (Math Z 228:569-588, 1998).This is obtained by constructing a new subring $S$ of $\operatorname{Hom}_R(\mathfrak{m},\mathfrak{m})$ and constructing enough torsion in $\Omega_S$, enabling us to pull back a nontrivial torsion to $\Omega_R$.