Singularities and Direct-sum Decompositions

Let (R; m;k) be a local ring (commutative and Noetherian). We will discuss existence and uniqueness of direct-sum decompositions of nitely generated R-modules. One says that R has nite CM type provided there are only nitely many indecomposable maximal Cohen-Macaulay R-modules up to isomorphism. Among complete equicharacteristic hypersurface rings with k algebraically closed of characteristic 6 = 2; 3; 5, the rings of nite CM type were characterized in a 1987 paper of Buchweitz, Greuel, Knn orrer and Schreyer. We will review this characterization and focus on recent eeorts to obtain a general classiication of local CM rings of nite CM type. If R is Henselian, one has the Krull-Schmidt uniqueness theorem for direct-sum decompo-sitions of nitely generated R-modules. For more general local rings, however, this uniqueness can fail dramatically. We analyze this failure by studying the monoid +(M) consisting of isomorphism classes of nitely generated modules that are direct summands of direct sums of copies of the nitely generated module M. It is not hard to see that +(M) is isomorphic to a monoid of the form G \N n , where G is a subgroup of Z n. Conversely, given any such monoid E, there exist a one-dimensional analytically unramiied local domain R and a nitely generated R-module M such that +(M) = E. Let (R; m;k) be a local (commutative, Noetherian) ring. The two main issues one must confront in studying direct-sum decompositions of nitely generated modules are to describe the indecomposable modules and to describe the diierent ways a module can be decomposed into a direct sum of indecomposable modules. In the rst part of this note we discuss the rst issue, focusing on the question of nite Cohen-Macaulay type. That is, which rings R (usually assumed to be Cohen-Macaulay) have, up to isomorphism, only nitely many indecomposable maximal Cohen-Macaulay modules? The second part of the paper deals with the second issue. Given a nitely generated module M we form the monoid +(M) consisting of isomorphism classes of nitely generated modules that are direct summands of direct sums of copies of M (with as the operation).