Cohen-Macaulay modules on hypersurface singularities

Summary

In the last few years the maximal Cohen-Macaulay modules over local rings of singularities have been studied by methods of representation theory, commutative algebra, and algebraic geometry. The article of M. Auslander in this volume and the present article are intended as a survey over some of the recent progress in this subject. This part of the survey will be concerned almost exclusively with the situation of complex hypersurface singularities. In fact it is centered around the following result from [Buchweitz-Greuel-Schreyer], [Knörrer]:

Theorem: Let R be the local ring of a (complex) hypersurface singularity. There are finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules over R if and only if R is the local ring of a simple singularity.

In the first part I want to give a brief account of the role the simple singularities play in the classification of hypersurface singularities. Of course this can be only a very rough and subjective sketch; a much broader and more competent description can be found e.g. in [Arnold], [Arnold et al.], [Durfee], [Slodowy]. The second chapter mainly reports on the group-theoretic and algebro-geometric description of maximal Cohen-Macaulay modules over two-dimensional simple hypersurface singularities. Chapter 3 contains a brief sketch of the proof of the theorem stated above.

In preparing these notes I profited very much from expositions of this and related material which R. Buchweitz has given at conferences in Göttingen and Oberwolfach.