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

TL;DR

This paper investigates the structure of maximal Cohen-Macaulay modules over certain one-dimensional rings, revealing conditions under which indecomposable modules of various ranks exist, thereby illuminating the monoid of module classes.

Contribution

It establishes the existence of indecomposable maximal Cohen-Macaulay modules of specified ranks over rings with infinite Cohen-Macaulay type, advancing understanding of their direct-sum behavior.

Findings

Existence of indecomposable modules of rank (r_1,...,r_s) when r_1 ≥ r_i.

Description of the monoid C(R) for rings with all branches of infinite Cohen-Macaulay type.

Conditions under which the monoid structure encodes non-uniqueness of module decompositions.

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 is greater than or equal to r_i for all i. This result allows us to describe the monoid C(R) when all analytic branches of R have infinite Cohen-Macaulay.