Zum Torsionsproblem bei Kurvensingularitäten. (The torsion problem on curve singularities)
type: Item
instance of: scholarly article
title: Zum Torsionsproblem bei Kurvensingularitäten. (The torsion problem on curve singularities)
zbMATH Open document ID: 0728.13011
DOI: 10.1007/BF01188678
author: Klaus Güttes
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: As an analogous conjecture for analytic algebras of the famous conjecture of Berger, the following conjecture is studied: ``Let \(R=kx_ 1,...,x_ n/I\) 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)\(\leq 13\).
Mathematics Subject Classification ID: 13J07
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 32B05
zbMATH DE Number: 4202437
zbMATH Keywords: Berger conjecture
zbMATH Keywords: analytic algebras
zbMATH Keywords: universally finite module of differentials
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
cites work: Differentialmoduln eindimensionaler lokaler Ringe
cites work: A note on the Hilbertfunction of a one-dimensional Cohen-Macaulay ring
cites work: Q5597625
cites work: Torsion des Differentialmoduls und Kotangentenmodul von Kurvensingularitäten
full work available at URL: https://doi.org/10.1007/bf01188678
OpenAlex ID: W1980035811
description: scientific article; zbMATH DE number 4202437