On Berger's conjecture about one dimensional local rings
type: Item
instance of: scholarly article
title: On Berger's conjecture about one dimensional local rings
zbMATH Open document ID: 0763.13006
DOI: 10.1007/BF01246739
author: Satoru Isogawa
published in: Archiv der Mathematik
publication date: 1993-02-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 13C12
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 124273
zbMATH Keywords: algebroid curve
zbMATH Keywords: differential module
zbMATH Keywords: nontrivial torsion
zbMATH Keywords: value semigroup
Wikidata QID: Q123365935
reviewed by: Robert W. Berger
MaRDI profile type: Publication
cites work: Differentialmoduln eindimensionaler lokaler Ringe
cites work: Zum Torsionsproblem bei Kurvensingularitäten. (The torsion problem on curve singularities)
cites work: Cotangent functors of curve singularities
cites work: Q5613258
cites work: Q3744273
description: scientific article; zbMATH DE number 124273