The Cohen-Macaulay type of Cohen-Macaulay rings

Abstract

Let R be an integrally closed complete local commutative noetherian Cohen-Macaulay domain with maximal ideal m and R/m= k an algebraically closed field. We abbreviate Cohen-Macaulay to CM and denote by CM(R) the category of finitely generated CM modules. In this paper we deal with the question of when R is of finite CM type, that is, has only a linite number of indecomposable CM modules. If dim R = 2, and R is a C-algebra with R/m= @, the complex numbers, then the R of finite CM type are exactly the fixed rings @[[X, Y]]‘, where X, Y are indeterminates and G c GL(2, C) is a finite group acting on @[[X, Y]] [ 17, 3, 151. For hypersurfaces in characteristic zero the R of finite CM type are exactly the simple hypersurface singularities in the sense of Arnold [18, 133, so in this case there is a nice connection with algebraic geometry. When dim R > 3 and R is not a hypersurface, we know only two examples of finite CM type, and they both have dimension 3. These are R,=kCCXo,X,,X,, Y,, Y~IlIWo~,-~, xoY,-x,Y,, x,Y,-x,Yc,) and R2 = k[ [X, Y, Z]lz*, where the generator of Z, acts by sending each variable to its negative, and the characteristic of k is different from 2. Each of these rings belongs naturally to a larger class. R, is a scroll of type (2, 1). We define more generally a scroll of type (m,, . . . . m,) (see [ 141) and show that when dim R 2 3, then all other scrolls (except type (1, l), which is a hypersurface) are of infinite CM type. R, is among the rings kCCJ’,, . . . . XJIG, where the order of G c GL(n, k) is invertible in k and n 2 3, and is the only one of finite CM type. These are the main results of this paper. To prove that the scroll of type (2, 1) is of finite CM type we use the 1 OOOl-8708/89 $7.50