Valuations and nonzero torsion in module of differentials

Abstract

Let (R,mR,k) be a one-dimensional complete local reduced k-algebra over a field of characteristic zero. Berger conjectured that R is regular if and only if the universally finite module of differentials ΩR 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 mR that is contained within the conductor ideal. This also generalizes a previous result of [7].