Valuations and Nonzero Torsion in Module of Differentials

Let $(R,{\mathfrak{m}}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials ${\mathrm{\Omega }}_R$ is torsion free. When $R$ is a domain, we prove the conjecture in several cases. Our techniques are primarily reliant on making use of the valuation semi-group of $R$. First, we establish a method of verifying the conjecture by analyzing the valuation semi-group of $R$ and orders of units of the integral closure of $R$. We also prove the conjecture in the case when certain monomials are missing from the monomial support of the defining ideal of $R$. These monomials are based on the smallest power of ${\mathfrak{m}}_R$ that is contained within the conductor ideal. This also generalizes a previous result of Cortiñas, Geller and Weibel.