CM-simple modules

(joint with N. Karroum) We are interested in the classification of maximal CM(=Cohen-Macaulay) modules on complex analytic singularites $(X,x)$, i.e. CM-modules over the ring $A={\mathcal O}_{X,x}$. By well known results of Buchweitz, Greuel, Knörrer and Schreyer a hypersurface singularity is simple in the sense of Arnold if and only if it admits only a finite number of indecomposable CM-modules. In this talk we are interested in CM-simple modules, that is CM-modules over $A$ that do not have non-trivial CM quotients. Clearly such modules form a subclass of the indecomposable ones.

It is easy to see that every CM module admits a filtration such that the successive quotients are
CM-simple. However we show that the associated graded object is not unique, in general. In this talk we classify all CM-simple modules over simple singularities and over minimally elliptic singularities. As follows from results of Yoshino and Dieterich the indecomposable modules over isolated singularities of a given rank form a bounded family hence this is also true for CM-simple modules. Here we show that for non isolated homogeneous singularities the set of CM-simple modules of a given rank is as well bounded. It is an open problem whether this remains true without the hypothesis that $A$ is homogeneous.

As an application we show that for homogeneous singularities with a discrete divisor class group the set of classes in $Cl(A)$ that are CM is a finite set. This gives a partial answer of a question raised by M. Hochster.