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Quasi-Gorenstein Modules

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Date Issued:
2018
Abstract/Description:
This thesis will study the various roles that quasi-Gorenstein modules and their properties play in the study of homological dimensions and linkage of modules. To that effect we begin by studying these modules in their own right. An $R$-module $M$ of grade $g$ will be quasi-Gorenstein if $\Ext_R^i(M,R)=0$ for $i\neq g$ and there is an isomorphism $M\cong\Ext_R^g(M,R)$. Such modules have many nice properties which we will explore throughout this thesis. We will show they help extend a characterization of diagonalizable matrices over principal ideal domains to more general rings. We will use their properties to help lay a foundation for a study of homological dimensions, helping to generalize the concept of Gorenstein dimension to modules of larger grade and present a connection to these new dimensions with certain generalized Serre conditions.We then give a categorical construction to the concept of linkage. The main motivation of such a construction is to generalize ideal and module linkage into one unified theory. By using the defintion of linkage presented by Nagel \cite{NagelLiaison}, we can use categorical language to define linkage between categories. One of the focuses of this thesis is to show that the history of linkage has been wrought with a misunderstanding of which classes of objects to study. We give very compelling evidence to suggest that linkage is a tool to gain information about the even linkage classes of objects. Further, scattered among the literature is a wide array of results pertaining to module linkage, homological dimensions, duality, and adjoint functor pairs and for which we show that these fall under the umbrella of this unified theory. This leads to an intimate relationship between associated homological dimensions and the linkage of objects in a category. We will give many applications of the theory to modules allowing one to cover vast grounds from Gorenstein dimensions to Auslander and Bass classes to local cohomology and local homology. Each of these gives useful insight into certain classes of modules by applying this categorical approach to linkage.
Title: Quasi-Gorenstein Modules.
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Name(s): York, Alexander, Author
Brennan, Joseph, Committee Chair
Martin, Heath, Committee Member
Ismail, Mourad, Committee Member
Kuebler, Stephen, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2018
Publisher: University of Central Florida
Language(s): English
Abstract/Description: This thesis will study the various roles that quasi-Gorenstein modules and their properties play in the study of homological dimensions and linkage of modules. To that effect we begin by studying these modules in their own right. An $R$-module $M$ of grade $g$ will be quasi-Gorenstein if $\Ext_R^i(M,R)=0$ for $i\neq g$ and there is an isomorphism $M\cong\Ext_R^g(M,R)$. Such modules have many nice properties which we will explore throughout this thesis. We will show they help extend a characterization of diagonalizable matrices over principal ideal domains to more general rings. We will use their properties to help lay a foundation for a study of homological dimensions, helping to generalize the concept of Gorenstein dimension to modules of larger grade and present a connection to these new dimensions with certain generalized Serre conditions.We then give a categorical construction to the concept of linkage. The main motivation of such a construction is to generalize ideal and module linkage into one unified theory. By using the defintion of linkage presented by Nagel \cite{NagelLiaison}, we can use categorical language to define linkage between categories. One of the focuses of this thesis is to show that the history of linkage has been wrought with a misunderstanding of which classes of objects to study. We give very compelling evidence to suggest that linkage is a tool to gain information about the even linkage classes of objects. Further, scattered among the literature is a wide array of results pertaining to module linkage, homological dimensions, duality, and adjoint functor pairs and for which we show that these fall under the umbrella of this unified theory. This leads to an intimate relationship between associated homological dimensions and the linkage of objects in a category. We will give many applications of the theory to modules allowing one to cover vast grounds from Gorenstein dimensions to Auslander and Bass classes to local cohomology and local homology. Each of these gives useful insight into certain classes of modules by applying this categorical approach to linkage.
Identifier: CFE0007268 (IID), ucf:52202 (fedora)
Note(s): 2018-08-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): Commutative Algebra -- Homological Algebra -- Homlogical Dimensions -- Ideal Linkage -- Module Linkage -- Linkage -- Filtrations -- Structure Theorems -- Quasi-Gorenstein Modules -- Module Theory -- Category Theory -- Homological Categories -- Categories with Linkage -- Adjoint Functors -- Auslander and Bass Classes -- Gorenstein Dimensions -- Noetherian Dimensions -- Width -- Grade -- Depth -- Auslander-Buchsbaum formula -- Auslander-Bridger formula -- Local Cohomology -- Local Homology -- Cohen-Macaulay rings -- Gorenstein rings
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0007268
Restrictions on Access: campus 2019-08-15
Host Institution: UCF

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