Zum Torsionsproblem bei Kurvensingularitäten. (The torsion problem on curve singularities) (Q804649)

As an analogous conjecture for analytic algebras of the famous conjecture of Berger, the following conjecture is studied: ``Let \(R=k[[x_ 1,...,x_ n]]/I\) be a reduced, one-dimensional analytic algebra over the perfect field k, such that the universally finite module of differentials D(R/k) is torsion-free. Then R is regular''. Let k be a field of characteristic zero. It is shown that the conjecture is true if there is a superficial element x in R such that \[ m(R)\leq \left( \begin{matrix} e\dim (R)\\ 2\end{matrix} \right)+\mu_{R/xR}({\mathfrak m}^ 2_{R/Rx})+\mu_{R/Rx}({\mathfrak m}^ 3_{R/Rx})+1, \] where m(R) is the multiplicity of R and \(\mu\) (M) is the minimal number of generators of a module M. This is also shown to be true if R is a domain such that m(R)\(\leq 9\) or a Gorenstein domain with m(R)\(\leq 13\).