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Auslander-Reiten theory for systems of submodule embeddings

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Date Issued:
2009
Summary:
In this dissertation, we will investigate aspects of Auslander-Reiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute Auslander-Reiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this situation of the Auslander and Ringel-Tachikawa Theorem which states that for an artinian ring R of finite representation type, each R-module is a direct sum of finite-length indecomposable R-modules. In cases where this applies, the indecomposable objects obtained in the Auslander-Reiten quiver give the building blocks for the objects in the category. We also briefly discuss in which cases systems of submodule embeddings form a Frobenius category, and for a few examples explore pointwise Calabi-Yau dimension of such a category.
Title: Auslander-Reiten theory for systems of submodule embeddings.
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Name(s): Moore, Audrey.
Charles E. Schmidt College of Science
Department of Mathematical Sciences
Type of Resource: text
Genre: Electronic Thesis Or Dissertation
Date Issued: 2009
Publisher: Florida Atlantic University
Physical Form: electronic
Extent: xi, 112 p. : ill.
Language(s): English
Summary: In this dissertation, we will investigate aspects of Auslander-Reiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute Auslander-Reiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this situation of the Auslander and Ringel-Tachikawa Theorem which states that for an artinian ring R of finite representation type, each R-module is a direct sum of finite-length indecomposable R-modules. In cases where this applies, the indecomposable objects obtained in the Auslander-Reiten quiver give the building blocks for the objects in the category. We also briefly discuss in which cases systems of submodule embeddings form a Frobenius category, and for a few examples explore pointwise Calabi-Yau dimension of such a category.
Identifier: 422624278 (oclc), 210496 (digitool), FADT210496 (IID), fau:3409 (fedora)
Note(s): by Audrey Moore.
Thesis (Ph.D.)--Florida Atlantic University, 2009.
Includes bibliography.
Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.
Subject(s): Artin algebras
Rings (Algebra)
Representation of algebras
Embeddings (Mathematics)
Linear algebraic groups
Persistent Link to This Record: http://purl.flvc.org/fcla/dt/210496
Use and Reproduction: http://rightsstatements.org/vocab/InC/1.0/
Owner Institution: FAU