One-dimensional local rings of infinite Cohen-Macaulay type

Abstract

Given a commutative ring R and a class [special characters omitted] of R-modules closed under isomorphism, finite direct sums and direct summands, one can ask whether every module in [special characters omitted] decomposes uniquely as a direct sum of indecomposable modules in [special characters omitted]. We restrict our attention to one-dimensional analytically unramified local rings (R, [special characters omitted], 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 [special characters omitted](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 [special characters omitted](R). (The rank of an R-module M is the tuple consisting of the vector-space dimensions of M_P over R_P, 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 [special characters omitted](R) when Rˆ/Q has infinite Cohen-Macaulay type for all minimal prime ideals Q of the [special characters omitted]-adic completion Rˆ of R.