On Cohen—Macaulay Modules on Surface Singularities

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Dedicated to V. I. Arnold Abstract. We study Cohen-Macaulay modules over normal surface singularities. Using the method of Kahn and extending it to families of modules, we classify Cohen-Macaulay modules over cusp singularities and prove that a minimally elliptic singularity is Cohen-Macaulay tame if and only if it is either simple elliptic or cusp. As a corollary, we obtain a classification of Cohen-Macaulay modules over log-canonical surface singularities and hypersurface singularities of type Tpqr; especially they are Cohen-Macaulay tame. We also calculate the Auslander-Reiten quiver of the category of Cohen-Macaulay modules in the considered cases. 2000 Math. Subj. Class. Primary 13C14, 13C05; Secondary: 16G50, 14J17.