One-dimensional local rings of infinite Cohen-Macaulay type

Given a commutative ring R and a class C of R-modules closed under isomorphism, finite direct sums and direct summands, one can ask whether every module in C decomposes uniquely as a direct sum of indecomposable modules in C . We restrict our attention to one-dimensional analytically unramified local rings (R, m , k) and to the class of maximal Cohen-Macaulay R-modules (i.e., non-zero finitely generated torsion-free R-modules). This class of modules has been studied when R has finite Cohen-Macaulay type---that is, there are only finitely many indecomposable maximal Cohen-Macaulay R-modules, up to isomorphism. In this dissertation, we study this class when R has infinite Cohen-Macaulay type.;One approach to the study of direct-sum decompositions over R is to describe the monoid C (R) of isomorphism classes of maximal Cohen-Macaulay R-modules with operation given by direct sum. The notion of rank of a module plays a fundamental role in describing the monoid C (R). (The rank of an R-module M is the tuple consisting of the vector-space dimensions of MP over RP, where P ranges over the minimal prime ideals of R.) We study which tuples occur as ranks of indecomposable maximal Cohen-Macaulay R-modules when there is at least one minimal prime ideal P of R such that R/P has infinite Cohen-Macaulay type. Based on these results, we give a precise description of the monoid C (R) when Rˆ/Q has infinite Cohen-Macaulay type for all minimal prime ideals Q of the m -adic completion Rˆ of R.