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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].

MSC:
14H20 Singularities of curves, local rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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References:
[1] Arnol’d, V.I.: Critical points of smooth functions. Proc. Int. Congress Math. Vancouver 1974, Vol. 1, 19-39
[2] Barth, W., Peters, Chr., Van de Ven, A.: Compact complex surfaces. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0718.14023
[3] Dietrich, E., Wiedemann, A.: The Auslander-Reiten quiver of a simple curve singularity. Preprint 1984
[4] Durfee, A.: Fifteen characterizations of rational double points and simple critical points. L’Enseignement Math.25, 131-163 (1979) · Zbl 0418.14020
[5] Green, E., Reiner, I.: Integral representations and diagrams. Mich. Math. J.25, 53-84 (1978) · Zbl 0365.16015
[6] Herzog, J.: Ringe mit nur endlich vielen Isomorphieklassen von maximalen unzerlegbaren Cohen-Macaulay-Moduln. Math. Ann.233, 21-34 (1978) · Zbl 0358.13009
[7] Jacobinsky, H.: Sur les ordres commutatifs avec un nombre fini de résaux indécomposables. Acta Math.118, 1-31 (1967) · Zbl 0156.04501
[8] Kirby, D., Tavallee, H.: On Cohen-Macaulay local rings of dimension one and embedding dimension two. Preprint, Southampton, Great Britain
[9] Knörrer, H.: The indecomposable Cohen-Macaulay modules over simple hypersurface singularities (in Vorbereitung) · Zbl 0617.14033
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