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Representation theory of one -dimensional local rings of finite Cohen -Macaulay type
Let (R, m , k) be a one-dimensional local ring. A non-zero R-module M is maximal Cohen-Macaulay (MCM) provided it is finitely generated and m contains a non-zerodivisor on M. In particular, R is a Cohen-Macaulay (CM) ring if R is a MCM module over itself. The ring R is said to have finite Cohen-Macaulay type (FCMT) if there are, up to isomorphism, only finitely many indecomposable MCM R-modules. The Krull-Schmidt Property is said to hold for a class C of R-modules provided whenever M1⊕&cdots;⊕Ms≅N1 ⊕&cdots;⊕Nt with Mi, Nj indecomposable modules in C , we have s = t and, after a possible reordering, Mi ≅ Nj. ^ This dissertation investigates whether or not the Krull-Schmidt property holds for the classes M (R) of all finitely generated R-modules and C (R) of MCM modules over rings with FCMT. In Chapter 2 we deal with the complete rings, where the Krull-Schmidt property is known to hold for all finitely generated modules. In this chapter we classify all indecomposable MCM modules. In Chapter 3 we give a classification of MCM modules over the non-complete rings. We are then able to determine when the Krull-Schmidt property holds for C (R) and M (R) and when we have the weaker property that any two representations of a MCM module as a direct sum of indecomposables have the same number of indecomposable summands. ^
Baeth, Nicholas R, "Representation theory of one -dimensional local rings of finite Cohen -Macaulay type" (2005). ETD collection for University of Nebraska - Lincoln. AAI3176768.