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Representation theory of the diagram An over the ring k[[x]]

Fix R = k[[x]]. Let Q<sub>n</sub> be the category whose objects are ((M₁,...,M<sub>n</sub>),(f₁,...,f<sub>n-1</sub>)) where each M<sub>i</sub> is a free R-module and f<sub>i</sub>:M<sub>i</sub>⟶M<sub>i+1</sub> for each i=1,...,n-1, and in which the morphisms are the obvious ones. Let β<sub>n</sub> be the full subcategory of Ω<sub>n</sub> in which each map f<sub>i</sub> is a monomorphism whose cokernel is a torsion module. It is shown that there is a full dense functor Ω<sub>n</sub>⟶β<sub>n</sub>. If X is an object of β<sub>n</sub>, we say that X <u>diagonalizes</u> if X is isomorphic to a direct sum of objects ((M₁,...,M<sub>n</sub>),(f₁,...,f<sub>n-1</sub>)) in which each M<sub>i</sub> is of rank one. We establish an algorithm which diagonalizes any diagonalizable object X of β<sub>n</sub>, and which fails only in case X is not diagonalizable.

Let Λ be an artin algebra of finite type. We prove that for a fixed C in mod(Λ) there are only finitely many modules A in mod(Λ) (up to isomorphism) for which a short exact sequence of the form 0⟶A⟶B⟶C⟶0 is indecomposable. / Ph. D. / incomplete_metadata

Identiferoai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/50001
Date January 1986
CreatorsCorwin, Stephen P.
ContributorsMathematics, Green, Edward, Farkas, Daniel R., Wheeler, Robert, Aull, Charles E., Brown, E.A.
PublisherVirginia Polytechnic Institute and State University
Source SetsVirginia Tech Theses and Dissertation
Detected LanguageEnglish
TypeDissertation, Text
Formatv, 65 leaves, application/pdf, application/pdf
RightsIn Copyright, http://rightsstatements.org/vocab/InC/1.0/
RelationOCLC# 14701354

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