On Auslander modules of normal surface singularities
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)].
scientific article; zbMATH DE number 4163061
scientific article