Bounds for indecomposable torsion-free modules

Abstract

Let M be a finitely generated torsion-free module over a one-dimensional reduced Noetherian ring R with finitely generated normalization. The rank of M is the tuple of vector-space dimensions of M P over each field R P ( R localized at P ), where P ranges over the minimal prime ideals of R . We assume that there exists a bound N R on the ranks of all indecomposable finitely generated torsion-free R -modules. For such rings, what bounds and ranks occur? Partial answers to this question have been given by a plethora of authors over the past forty years. In this article we provide a final answer by giving a concise list of the ranks of indecomposable modules for R a local ring with no condition on the characteristic. We conclude that if the rank of an indecomposable module M is ( r , r , … , r ) , then r ∈ { 1 , 2 , 3 , 4 , 6 } , even when R is not local.