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Subfunctors of Extension Functors University of Kentucky UKnowledge Theses and Dissertations--Mathematics Mathematics 2014 Subfunctors of Extension Functors Furuzan Ozbek University of Kentucky, [email protected] Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Ozbek, Furuzan, "Subfunctors of Extension Functors" (2014). Theses and Dissertations--Mathematics. 16. https://uknowledge.uky.edu/math_etds/16 This Doctoral Dissertation is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Mathematics by an authorized administrator of UKnowledge. For more information, please contact [email protected]. STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained needed written permission statement(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine) which will be submitted to UKnowledge as Additional File. I hereby grant to The University of Kentucky and its agents the irrevocable, non-exclusive, and royalty-free license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless an embargo applies. I retain all other ownership rights to the copyright of my work. I also retain the right to use in future works (such as articles or books) all or part of my work. I understand that I am free to register the copyright to my work. REVIEW, APPROVAL AND ACCEPTANCE The document mentioned above has been reviewed and accepted by the student’s advisor, on behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of the program; we verify that this is the final, approved version of the student’s thesis including all changes required by the advisory committee. The undersigned agree to abide by the statements above. Furuzan Ozbek, Student Dr. Edgar Enochs, Major Professor Dr. Peter Perry, Director of Graduate Studies Subfunctors of Extension Functors DISSERTATION Adissertationsubmittedinpartial fulfillment of the requirements for the degree of Doctor of Philosophy in the College of Arts and Sciences at the University of Kentucky By Furuzan Ozbek Lexington, Kentucky Director: Dr. Edgar Enochs, Professor of Mathematics Lexington, Kentucky 2014 c Copyright Furuzan Ozbek 2014 ABSTRACT OF DISSERTATION Subfunctors of Extension Functors This dissertation examines subfunctors of Ext relative to covering (enveloping) classes and the theory of covering (enveloping) ideals. The notion of covers and envelopes by modules was introduced independently by Auslander-Smalø and Enochs and has proven to be beneficial for module theory as well as for representation theory. The first few chapters examine the subfunctors of Ext and their properties. It is showed how the class of precoverings give us subfunctors of Ext. Furthermore, the characterization of these subfunctors and some examples are given. In the latter chapters ideals, the subfunctors of Hom, are investigated. The definition of cover and envelope carry over to the ideals naturally. Classical conditions for existence theorems for covers led to similar approaches in the ideal case. Even though some theorems such as Salce’s Lemma were proven to extend to ideals, most of the theorems do not directly apply to the new case. It is showed how Eklof & Trlifaj’s result can partially be extended to the ideals generated by a set. In that case, one also obtains a significant result about the orthogonal complement of the ideal. We relate the existence theorems for covering ideals of morphisms by identifying the morphisms with objects in A2 and obtain a sufficient condition for the existence of covering ideals in a more general setting. We finish with applying this result to the class of phantom morphisms. KEYWORDS: Homological Algebra, Cover, Subfunctor of Ext, Covering ideal, Phantom morphism Author’s signature: Furuzan Ozbek Date: May 4, 2014 Subfunctors of Extension Functors By Furuzan Ozbek Director of Dissertation: Edgar Enochs Director of Graduate Studies: Peter Perry Date: May 4, 2014 To my family for their love and endless support ACKNOWLEDGMENTS Completion of this doctoral dissertation was possible with the support of several peo- ple. Foremost, I am extremely grateful to my advisor, Professor Edgar Enochs, for his valuable guidance, scholarly inputs and consistent encouragement I received throughout my research work. Professor Enochs has always made himself available to answer my questions, clarify my doubts and I consider it as a great opportunity to do my doctoral programme under his guidance and to learn from his research expertise. This dissertation would not be possible without his patience and encouragement. I would like to thank my committee members, Professor Alberto Corso, Professor Uwe Nagel, Professor Ruriko Yoshida for their encouragement and insightful comments. My sincere thanks also goes to Professor Sergio Estrada and Professor Pedro A. Guil who gave me a chance to be a part of their research group and were kind enough to support me during my visit at Universidad de Murcia. Last but not the least, I would like to thank my family: my parents Emine Ozbek and Mustafa Ozbek, my companion Chad Landers and my dear friend Sema Gunturkun. They helped me keep going even when it seemed impossible and cheered me up when I was feeling down. Thank you all for being there throughout this hectic journey. iii CONTENTS Acknowledgments..................................... iii Contents.......................................... iv Chapter1 INTRODUCTION ............................. 1 Chapter 2 SUBFUNCTORS OF Ext ......................... 3 2.1 Preliminaries . 3 2.2 Subfunctorscorrespondingtoaprecoveringclass . 5 2.3 Results on Ext ................................ 9 P 2.4 Characterization of Ext1 ........................... 11 P Chapter 3 ABSOLUTELY PURE MODULES AND THE SUBFUNCTOR Axt .. 16 3.1 On absolutely pure modules . 16 3.2 The subfunctor AxtR for coherent rings . 20 Chapter 4 PHANTOM MORPHISM RELATIVE TO A SUBFUNCTOR . 22 4.1 Properties of phantom morphisms . 22 4.2 Phantom morphisms relative to Axt ..................... 26 Chapter5 IDEALSGENERATEDBYASET. 28 5.1 Preliminaries . 28 5.2 Properties of an ideal generated by a set . 29 5.3 Main result . 38 5.4 Idealsclosedundersums ............................ 39 5.5 Ideals generated by a set in the extended sense . 41 Chapter6 IDEALSCLOSEDUNDERDIRECTLIMITS . 43 6.1 The category 2 ................................ 43 6.2 A Sufficient ConditionA for Covering Ideals . 44 6.3 FilteringPhantomMorphisms . 47 Bibliography . 53 Vita ............................................ 56 EDUCATION..................................... 56 PUBLICATIONS................................... 56 AWARDS ....................................... 56 iv Chapter 1 INTRODUCTION The theory of covers and envelopes was first introduced by Enochs in [16] for classes of injective modules and flat modules. Auslander-Smalø had also introduced essentially the same notion in [4] for modules over an Artinian algebra. The study of these notions have proven to be beneficial for module theory as well as representation theory (see, for example, [2], [8], [11], [22], [30]). With the help of these notions one can define resolutions for precovering (preenvelop- ing) classes and then look at the derived functors of Hom relative to these classes. In particular one can prove that these derived functors (denoted Ext1 )giveusasubfunctor of Ext1. The study of these relative derived functors have beenP considered in di↵erent settings (see [2], [12], [28], [45]) and have been useful for several areas such as tilting theory, finitistic dimension theory, representation theory of Artin algebras, etc. We will mostly focus on the subfunctors Ext1 relative to a precovering (preenveloping) class and formulate the characterization forP the short exact sequences in Ext1 . P One of the recent trends in the theory of covers (envelopes) that attractedP wide inter- est is the ideals. The concept of covers and envelopes carry over to the ideal case very naturally and hence leads to the statement of existence theorems for covers and envelopes associated to these ideals, similar to the ones obtained in the classical theory of covers and envelopes. Even though some theorems such as Salce’s Lemma were proven to extend to ideals (see [32]), most of the theorems do not directly apply to the new case. In [37] we showed how Eklof & Trlifaj’s result [15] can partially be extended to the ideals generated by a set. In that case, one also obtains a significant result about the orthogonal comple- ment of the ideal. In collaboration with Prof. Estrada and Prof. A. Guil (see [21]), we also have studied the conditions under which existence of covering ideals were guaranteed in a more general setting. Our approach was to relate the existence theorems for covering ideals of morphisms by identifying the morphisms with objects in a Grothendieck cate- gory 2 of all representations by left R-modules of the quiver A2 : .Thenonealso obtainsA a new and elementary proof for the existence of phantom covers•!• (cf. [31]). The theory of ideal approximation
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