Ranks of Indecomposable Modules over Rings of Infinite Cohen-Macaulay Type

Abstract

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(R) 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(R) 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 ≥ r i for all i ∈ {1,…, s}. This result allows us to describe the monoid ℭ(R) when has infinite Cohen-Macaulay type for every minimal prime ideal Q of the 𝔪-adic completion .