We have shown in the last chapter that Gorenstein rings of finite representation type are simple singularities. This chapter aims at showing that the converse is true for onedimensional local rings. More generally we are able to provide a necessary and sufficient condition for one dimensional local rings to be of finite representation type; see Theorem (9.2). Furthermore we can draw the AR quivers for simple singularities of dimension one.

Throughout the chapter (R, m, k) is a one-dimensional analytic local algebra over k, where k is an algebraically closed field of characteristic 0. Since we are interested in the finiteness of representation type, we always assume R has only an isolated singularity, or equivalently R is reduced; see (3.1) and (4.22). As before let ℭ(R) be the category of CM modules over R. Recall that the objects in ℭ(R) are exactly the modules without torsion, (1.5.2).

(9.1) DEFINITION. Let R* be the integral closure of R in its total quotient ring. Note that R* is also a one-dimensional (not necessarily local) ring which is finite over R. A local ring S is said to birationally dominateR if R ⊂ S ⊂ R*.

One of the aims of this chapter is to prove

(9.2) THEOREM. (Greuel-Knörrer) The following two conditions are equivalent:

(9.2.1) R is of finite representation type;

(9.2.2) R birationally dominates a simple curve singularity.

In particular we have:

(9.3) COROLLARY. A local ring of simple singularity of dimension one is of finite representation type.