On Berger's conjecture about one dimensional local rings
Let \(R\) be a complete one-dimensional local \(k\)-algebra over an algebraically closed field of characteristic zero. The conjecture is that if \(R\) is not regular then the universally finite differential module of \(R\) over \(k\) has nontrivial torsion. The author proves this conjecture in three special situations. (1) If \(n\) denotes the embedding dimension and \(e\) the multiplicity of \(R\), the conjecture is true for rings with \(n(n-1)\geq 2(e+1)\). This is a special case of a proposition of \textit{Güttes} with a simple proof. (2) The conjecture is true if \({\mathfrak d}^{n-1}(R/k):{\mathfrak d}^{n- 1}(R/k)\neq R:{\mathfrak d}^{n-1}(R/k)\), where \({\mathfrak d}^{n-1}(R/k)\) denotes the \((n-1)\)-st Kähler different of \(R/k\). (3) The author gives an expression of the length of the torsion in terms of the value semigroup of \(R\) from which he can show that the conjecture holds if there are no ``exceptional differential''. This proves a theorem of Yoshino the original proof of which was incomplete.
scientific article; zbMATH DE number 124273