Ranks of indecomposable modules over rings of infinite Cohen-Macaulay type

Let (R,m,k) be a one-dimensional analytically unramified local ring with minimal prime ideals P_1,…,P_s. Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by the monoid C® of isomorphism classes of maximal Cohen-Macaulay R-modules: the structure of this monoid reveals, for example, whether or not every maximal Cohen-Macaulay module is uniquely a direct sum of indecomposable modules; when uniqueness does not hold, invariants of this monoid give a measure of how badly this property fails. The key to understanding the monoid C® is determining the ranks of indecomposable maximal Cohen-Macaulay modules. Our main technical result shows that if R/P_1 has infinite Cohen-Macaulay type and the residue field k is infinite, then there exist |k| pairwise non-isomorphic indecomposable maximal Cohen-Macaulay R-modules of rank (r_1,…,r_s) provided r_1 is greater than or equal to r_i for all i. This result allows us to describe the monoid C® when all analytic branches of R have infinite Cohen-Macaulay.