Über den Derivationenmodul und das Jacobi-Ideal von Kurvensingularitäten
type: Item
instance of: scholarly article
title: Über den Derivationenmodul und das Jacobi-Ideal von Kurvensingularitäten
zbMATH Open document ID: 0543.14014
DOI: 10.1007/BF01163171
author: E. Kunz
author: R. Waldi
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173476
review text: Let \(R=K[X_ 1,...,X_ n]/I\) be a reduced curve singularity over an algebraically closed field K of characteristic 0. If R is Gorenstein the derivation module \(Der_ K(R)\) contains a canonical free submodule D of rank 1, the ''module of trivial derivations''. In case R is a complete intersection, i.e. \(I=(F_ 1,...,F_{n-1}),\) the derivation \(\delta:=\sum^{n}_{k=1}(-1)^{n-k}\Delta_ k(\partial /\partial X_ k),\) where \(\Delta_ k\) denote the maximal minors of the Jacobian \(\partial(F_ 1,...,F_{n-1})/\partial(X_ 1,...,X_ n),\) is a basis element of D. The following facts are proved: Theorem 1. The length of \(Der_ K(R)/D\) is \(\leq 2\delta_ R-r_ R+1,\) where \(\delta_ R\) is the degree of singularity of R and \(r_ R\) the number of branches of R. Equality holds iff R is a quasihomogeneous singularity. Theorem 2. The minimal number of generators of \(Der_ K(R)/D\) is r(coker c\({}_ R)\), where \(c_ R:\Omega^ 1_{R/K}\to \omega^ 1_{R/K}\) is the canonical homomorphism from Kähler differentials to regular differentials and r denotes the Cohen-Macaulay type of a module. Moreover, the following conditions are equivalent: (a) \(Der_ K(R)/D\) is a cyclic R-module, (b) R is quasihomogeneous. - Theorem 3. If R is a complete intersection and J denotes the Jacobian ideal of R/K then R is quasihomogeneous iff R/J is a Gorenstein ring.
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 3863352
zbMATH Keywords: number of generators of derivation module
zbMATH Keywords: length of derivation module
zbMATH Keywords: reduced curve singularity
zbMATH Keywords: complete intersection
MaRDI profile type: Publication
cites work: The Milnor number and deformations of complex curve singularities
cites work: Numerische Charakterisierung Quasihomogener GORENSTEIN-Kurvensingularitäten
cites work: Der kanonische Modul eines Cohen-Macaulay-Rings. (The canonical moduls of a Cohen-Macaulay-ring)
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cites work: Free Derivation Modules on Algebraic Varieties
cites work: Q4075559
description: scientific article; zbMATH DE number 3863352