On Auslander modules of normal surface singularities
type: Item
instance of: scholarly article
title: On Auslander modules of normal surface singularities
zbMATH Open document ID: 0708.14023
DOI: 10.1007/BF02568491
author: Kurt Behnke
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155468
review text: Let \(R\) be the local ring of a normal surface singularity, then a sequence \[ 0\rightarrow\omega\rightarrow A\rightarrow R\rightarrow k\rightarrow 0,\] corresponding to a nonzero element in \(\mathrm{Ext}^2_R(k,\omega)\) \((k=R/m\), \(\omega\) the dualizing module) is called the fundamental sequence of \(R\); the module \(A\) the ``Auslander module'' of \(R\). The above sequence determines in a certain sense the so called ``Auslander-Reiten quiver'' of \(R\). The article under review considers the following cases (always over \(\mathbb{C})\): (1) \(R\) is rational with reduced fundamental cycle (in the minimal resolution), or (2) \(R\) is minimally elliptic. Then \(A\cong (\Omega^1_R)\) if and only if \(X\) is quasihomogeneous. Under one of these conditions, this is the affirmative answer to a question of Martsinkovsky, whether this equivalence is true in general [cf. \textit{A. Martsinkovsky}, ``Almost split sequences and Zariski differentials'', Thesis (Brandeis Univ. 1987); see also Trans. Am. Math. Soc. 319, No. 1, 285--307 (1990; Zbl 0715.13019)].
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 4163061
zbMATH Keywords: normal surface singularity
zbMATH Keywords: fundamental sequence
zbMATH Keywords: Auslander module
zbMATH Keywords: Auslander-Reiten quiver
reviewed by: Marko Roczen
MaRDI profile type: Publication
cites work: Some Numerical Criteria for Contractability of Curves on Algebraic Surfaces
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cites work: Infinitesimal deformations of cusp singularities
cites work: On the module of Zariski differentials and infinitesimal deformations of cusp singularities
cites work: Reflexive modules on quotient surface singularities.
cites work: Lokale und globale Invarianten der Hilbertschen Modulgruppe
cites work: Rings mit only finitely many isomorphism classes of maximal indecomposable Cohen-Macaulay modules
cites work: Hilbert modular surfaces
cites work: The Zariski-Lipman conjecture in the graded case
cites work: Reflexive modules on minimally elliptic singularities
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description: scientific article; zbMATH DE number 4163061