Dokument: Proper Klassen von kurzen exakten Folgen und Strukturtheorie von Moduln
| Titel: | Proper Klassen von kurzen exakten Folgen und Strukturtheorie von Moduln | |||||||
| Weiterer Titel: | Proper classes of short exact sequences and structure theory of modules | |||||||
| URL für Lesezeichen: | https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=15658 | |||||||
| URN (NBN): | urn:nbn:de:hbz:061-20100720-103809-8 | |||||||
| Kollektion: | Dissertationen | |||||||
| Sprache: | Englisch | |||||||
| Dokumententyp: | Wissenschaftliche Abschlussarbeiten » Dissertation | |||||||
| Medientyp: | Text | |||||||
| Autor: | Preisser Montano, Carlos Federico [Autor] | |||||||
| Dateien: |
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| Beitragende: | Prof. Dr. Wisbauer, Robert [Gutachter] Prof. Dr. Schröer, Stefan [Gutachter] | |||||||
| Stichwörter: | Proper classes, relative homological algebra, torsion theory, cotorsion pairs | |||||||
| Dewey Dezimal-Klassifikation: | 500 Naturwissenschaften und Mathematik » 510 Mathematik | |||||||
| Beschreibung: | In the late 50's Buchsbaum imposed some axioms to a class of monomorphisms in order to define the functors $\ExtPn$ in an exact category without using injectives or projectives. This axioms were reformulated in Mac Lane's "Homology" for a class $\PC$ of short exact sequences in an abelian category $\catC$. A class $\PC$ satisfying these axioms is called a proper class. The class of all short exact sequences ($\abs$) and the class of all splitting short exact sequences ($\Split$) in $\catC$ are examples of proper classes. A non-trivial example is the class of the pure exact sequences ($\Cohn$) in $\RMod$, i.e. the class of all short exact sequences such that the finitely presented modules are projective with respect to them. We say that the class of pure exact sequences is projectively generated by the class of finitely presented modu\-les. Proper classes injectively generated by a class of modules are defined dually. Later on Harrison noted that the inclusions of complement subgroups of abelian groups define a proper class. Generalov and Stenstr\"om generalized this to modules over arbitrary rings. Alizade and Mermut continued with this investigations. Recently Al-Takhman et al. introduced, for a radical $\tau$, the notion of $\tau$-supplements and proved that the inclusions of $\tau$-supplements define a proper class which coincide with the proper class injectively generated by the $\tau$-torsionfree modules.
Our first aim is to generalize the notions of supplements and complements by relating them to a class of modules $\C$ closed under submodules and factor modules. For example: the class of semisimple modules, the class of singular modules, the class of small modules, a hereditary pretorsion class, the class of noetherian modules, etc. We prove that the classes of short exact sequences determined by the inclusions of $\C$-supplements and $\C$-complements define proper classes. In particular for $\C=\RMod$ we recover the supplements and complements, for $\C=\{\textrm{singular modules}\}$ the $\delta$-supplements introduced by Zhou and for $\C=\{\textrm{simple modules}\}$ the $\Rad$-supplements and $\Soc$-complements. We show how this proper classes are related and we use them to characterize some classes of modules (e.g. GCO-modules). Associated with each proper class $\PC$ in an abelian category $\catC$ there are, among others, the classes: $\PC\textrm{-flats}=\{X\in \catC \mid \ExtC(X, C)=\ExtP(X, C)\, \forall\, C\in \catC\}$ and $\PC\textrm{-divisibles}=\{X\in \catC \mid \ExtC(C, X)=\ExtP(C, X)\, \forall\, C\in \catC\}$. We determine this classes for the proper classes of supplements (compl.), $\tau$-supplements ($\tau$-compl.) and $\C$-supplements ($\C$-compl.). Cotorsion pairs were introduced by Salce in the 70's for abelian groups. A cotorsion pair in an abelian category is a pair $(\cotor, \cotorfree)$ of classes of objects in $\catC$ which are orthogonal with respect to the functor $\ExtC$. For example the pairs: $(\textrm{projectives of } \catC, \catC)$ and $(\catC, \textrm{injectives of } \catC)$ are cotorsion pairs. Cotorsion pairs have gained interest in the last years in representation theory, in particular they were used to prove the flat cover conjeture. We define an order-reversing correspondence $\PC\mapsto (\PC\textrm{-flats}, \PC\textrm{-flats}^{\perp})$ between proper classes and cotorsion pairs which preserves arbitrary meets. This correspondence is bijective if we restrict it to the class of those injectively generated proper classes $\PC$ such that $\PC$-injectives $=$ $\PC$-flats$^{\perp}$ (Xu proper classes). We show how properties of proper classes yield properties of cotorsion pairs. Given two proper classes $\PC$ and $\R$ in an abelian category we define the classes: $\PC$-$\R$-flats $=\{X\in \catC \mid \ExtP(X, C)\subseteq \ExtR(X, C)\, \forall\, C\in \catC\}$, $\PC$-$\R$-divisibles $=\{X\in \catC \mid \ExtP(C, X)\subseteq \ExtR(C, X)\, \forall\, C\in \catC\}$ and $\PC$-$\R$-regulars which are those $X'$s in $\catC$ such that every short exact sequence in $\PC$ with middle term $X$ belongs to $\R$. This notions cover several concepts of module theory: projective modules $=$ $\abs$-$\Split$-flats, flat modules $=$ $\abs$-$\Cohn$-flats, injective modules $=$ $\abs$-$\Split$-divisibles, absolutely pure modules $=$ $\abs$-$\Cohn$-divisibles, extending modules $=$ $\Compl$-$\Split$-regulars, lifting modules $=$ $\Suppl$-$\Split$-regulars which are amply supplemented, regular modules $=$ $\abs$-$\Cohn$-regulars. \nopagebreak Finally we discuss cotorsion pairs relative to a proper class $\PC$ as introduced by Hovey in connection with model categories in the sense of Quillen. They are orthogonal pairs $(\cotor, \cotorfree)$ of classes of objects with respect to the functor $\ExtP$ instead of $\ExtC$. We express some results on relative approximations (covers and envelopes) due to Auslander and Solberg in terms of proper classes. | |||||||
| Lizenz: |  Urheberrechtsschutz | |||||||
| Fachbereich / Einrichtung: | Mathematisch- Naturwissenschaftliche Fakultät » WE Mathematik » Algebra und Zahlentheorie | |||||||
| Dokument erstellt am: | 20.07.2010 | |||||||
| Dateien geändert am: | 16.07.2010 | |||||||
| Promotionsantrag am: | 17.05.2010 | |||||||
| Datum der Promotion: | 01.07.2010 |
