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Einfache Kurvensingularitäten und torsionsfreie Moduln. (German) Zbl 0553.14011
Let R be the complete local ring of a reduced curve singularity with algebraically closed residue field of characteristic 0. R is said to have finite Cohen-Macaulay (CM) type if there are only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules. The main result of the paper is that R has finite CM type if it dominates the local ring \(R'\) of a simple plane curve singularity in the sense of Arnol’d (i.e. of type \(A_ k\), \(D_ k\) or \(E_ 6\), \(E_ 7\), \(E_ 8)\). That \(R\) dominates \(R'\) means just that \(R'\subset R\subset \tilde R',\) where \(\tilde R'=\) normalization of \(R'\). As a corollary it is shown that if R itself is a plane curve singularity, then R is of finite CM type if R is simple. This result remains true for curves in arbitrary characteristic [K. Kiyek and G. Steinke, ”Einfache Kurvensingularitäten in beliebiger Charakteristik”, Arch. Math. (to appear; see the following review)] and for higher dimensional hypersurface singularities [cf. M. Artin and J.-L. Verdier, Math. Ann. 270, 79-82 (1985; Zbl 0553.14001), J. Auslander, Herzog and H. Esnault, ”Reflexive modules on quotient surface singularities”, J. Reine Angew. Math. (to appear; Zbl 0553.14016) in dimension 2; H. Knörrer and R. Buchweitz, G.-M. Greuel and Schreyer (to appear) in higher dimensions].

14H20 Singularities of curves, local rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Full Text: DOI EuDML
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