Our main purpose in this paper is to explore some instances of almost split sequences occuring in algebraic geometry. We will be dealing mainly with three topics:

1. (1) The connections between the structure of the almost split sequences for reflexive modules over rational double points and the desingularization graph of the singularity;

2. (2) The existence of almost split sequences for coherent sheaves over nonsinguiar and Gorenstein projective curves;

3. (3) The question of which complete integrally closed Cohen-Macaulay local domains are of finite Cohen-Macaulay type, i.e. have, up to isomorphism only a finite number of indecomposable Cohen-Macaulay modules.

This paper is almost a verbatim account of the last two lectures I gave at the Durham symposium on the representation theory of algebras. Since these lectures were purely expository, no proofs are given.

§1. Complete rational double points.

Throughout this section k is an algebraically closed field. We recall that the complete rational double points over k can be described as follows. Let k[[u,v,w]] be the ring of formal power series in three variables u, v, w. Then the complete rational double points over k are k-algebras R of the form k[[u,v,w]]/(f(u,v,w)) with f(u,v,w) a 166 non-zero element in the maximal ideal (u,v,w) of k[[u,v,w]] such that R is an integrally closed domain with a finite class group.