A Krull–Schmidt theorem for one-dimensional rings of finite Cohen–Macaulay type

This paper determines when the Krull–Schmidt property holds for all finitely generated modules and for maximal Cohen–Macaulay modules over one-dimensional local rings with finite Cohen–Macaulay type. We classify all maximal Cohen–Macaulay modules over these rings, beginning with the complete rings where the Krull–Schmidt property is known to hold. We are then able to determine when the Krull–Schmidt property holds over the non-complete local rings and when we have the weaker property that any two representations of a maximal Cohen–Macaulay module as a direct sum of indecomposables have the same number of indecomposable summands.