In this dissertation, let B be a triangulated category and let D be an extension-closed subcategory of B. First, we give some new characterizations of an Auslander-Reiten triangle in D, which yields some necessary and sufficient conditions for D to have Auslander-Reiten triangles. Next, we study when an Auslander-Reiten triangle in B induces an Auslander-Reiten triangle in D. As an application, we study Auslander-Reiten triangles in a triangulated category with a t-structure. In case the t-structure has a t-hereditary heart, we establish the connection between the Auslander-Reiten triangles in B and the Auslander-Reiten sequences in the heart. Finally, we specialize to the bounded derived category of all modules of a noetherian algebra over a complete local noetherian commutative ring. Our result generalizes the corresponding result of Happel’s in the bounded derived category of finite dimensional modules of a finite dimensional algebra over an algebraically closed field.

Made available in DSpace on 2015-03-26T13:45:37Z (GMT). No. of bitstreams: 1 texto completo.pdf: 446590 bytes, checksum: 8c515244f3fa52b730a12770059cccea (MD5) Previous issue date: 2014-02-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this paper, we prove the existence of Auslander-Reiten triangles (TAR) for compact objects in triangulated categories compactly generated. The prove presented is an application of the theorem of Brown representability in derived categories for compact complex, ie, given Z be a compact and indecomposable complex, we show that there is a Auslander-Reiten triangle X->U->Y->v->Z->w->TX in K-b(^) which is equivalent to D(^), where ^ is a finite-dimensional k-algebra over an algebraically closed field. Furthermore, we have that a triangle Auslander-Reiten wihch start with the projective resolution of a indecomposable and non-injective module T-¹pM->alfa->Y->Beta->(pDM)*->y->pM induces an Auslander-Reiten sequence(SAR) 0->M->alfa¹->Cok¹ (Y)-> beta¹->Tr DM->0. How Mod(^) and D(^) are Krull-Schmidt, and classes of indecomposable objects and generators of irreducible morphisms of these categories occur in the SAR's and TAR's, respectively, these results provide us with a skillful tool to know the structures Mod(^) and D(^) of k-algebras. Moreover, we present examples using the representation theory of quivers of an algebra of paths. / Neste trabalho, apresentamos uma prova da existência de triângulos de Auslander-Reiten(TAR) para objetos compactos em categorias trianguladas compactamente geradas. A prova apresentada é uma aplicação do Teorema da Representabilidade de Brown em categorias derivadas para complexos compactos, ou seja, dado Z um complexo compacto e indecomponíveL mostramos que existe um triângulo X->U->Y->v->Z->w->TX de Auslander-Reiten em K-b(^) que é equivalente à Db(^), onde ^ é uma k-álgebra de dimensão finita sobre um corpo algébricamente fechado. Além disso, temos que um triângulo de Auslander-Reiten que começa com a resolução projetiva de um módulo indecomponível não-injetivo T-¹pM->alfa->Y->Beta->(pDM)*->y->pM induz uma sequência de Auslander-Reiten(SAR) 0->M->alfa¹->Cok¹ (Y)-> beta¹->Tr DM->0. Como MOd(^) e D(^) são Krull-Remak-Schmidt, e as classes de objetos inde- componíveis e os geradores de morfismos irredutíveis destas categorias ocorrem nas SAR's e nos TAR's, respectivamente, estes resultados nos fornecem uma hábil ferramenta para conhecer as estruturas de Mod(^) e D(^) de k-álgebras. Além disso, apresentamos exemplos utilizando a teoria de representação de quivers de uma álgebra de caminhos.

A complexidade de um módulo M, sobre uma álgebra de dimensão finita R, é a medida do crescimento da dimensão de suas sizigias. No nosso trabalho, estudamos esse conceito, nos concentrando muito mais no caso das álgebras autoinjetiva. Relacionamos esse crescimento com o comportamento da componente do carcás de Auslander-Reiten, a qual o módulo M pertence. Em particular, estudamos, com bastante cuidado, o caso em que a complexidade é 1, o que significa que a dimensão das sizigias são eventualmente constante. Surpreendentemente, o comportamento de todos os módulos numa mesma componente é muito parecido. / The complexity of a module M under a finite dimensional algebra R is the measure of the growth of its syzygies\' dimension. In our work, we study this concept concentrating on the case of the selfinjective algebras. We relate this growth with the behavior of the Auslander-Reiten component containing this module. In particular, we study, carefully, the case in which the complexity is 1. Surprisingly, the behavior of every module in the same component as M is very similar.

A complexidade de um módulo M, sobre uma álgebra de dimensão finita R, é a medida do crescimento da dimensão de suas sizigias. No nosso trabalho, estudamos esse conceito, nos concentrando muito mais no caso das álgebras autoinjetiva. Relacionamos esse crescimento com o comportamento da componente do carcás de Auslander-Reiten, a qual o módulo M pertence. Em particular, estudamos, com bastante cuidado, o caso em que a complexidade é 1, o que significa que a dimensão das sizigias são eventualmente constante. Surpreendentemente, o comportamento de todos os módulos numa mesma componente é muito parecido. / The complexity of a module M under a finite dimensional algebra R is the measure of the growth of its syzygies\' dimension. In our work, we study this concept concentrating on the case of the selfinjective algebras. We relate this growth with the behavior of the Auslander-Reiten component containing this module. In particular, we study, carefully, the case in which the complexity is 1. Surprisingly, the behavior of every module in the same component as M is very similar.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-16T11:59:48Z No. of bitstreams: 1 arquivototal.pdf: 1814472 bytes, checksum: 50307319d745b002230f3dadb2039246 (MD5) / Made available in DSpace on 2017-08-16T11:59:48Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1814472 bytes, checksum: 50307319d745b002230f3dadb2039246 (MD5) Previous issue date: 2016-08-09 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this dissertation, we study the so-called totally re°exive modules and the notion of Goren-stein dimension over Noetherian commutative rings. The main purpose is to prove the important Auslander-Bridger formula and the Gorenstein theorem, which will allow us to characterize Goren-stein local rings through total re°exivity, as well as to provide su±cient conditions for the property of G-regularity. We furnish, moreover, interesting examples and counterexamples. / Nesta disserta»c~ao, estudamos os chamados m¶odulos totalmente re°exivos e a no»c~ao de dimens~ao de Gorenstein sobre an¶eis comutativos Noetherianos. A principal ¯nalidade ¶e demonstrar a impor- tante f¶ormula de Auslander-Bridger e o Teorema de Gorenstein, o que permitir¶a caracterizar an¶eis locais Gorenstein atrav¶es de re°exividade total, bem como apresentar condi»c~oes su¯cientes para a propriedade de G-regularidade. Fornecemos, tamb¶em, exemplos e contra-exemplos interessantes.

This thesis consists of three parts. In the first part we look at Hopf algebras. We classify pointed rank one Hopf algebras over fields of prime characteristic which are generated as algebras by the first term of the coradical filtration. These Hopf algebras were classified by Radford and Krop for fields of characteristic zero. We obtain three types of Hopf algebras presented by generators and relations. The third type is new and has not previously appeared in literature. The second part of this thesis deals with Auslander-Reiten theory of finitedimensional algebras over fields. We consider G-transitive algebras and develop necessary conditions for them to have Auslander-Reiten components with Euclidean tree class. Thereby a result in [F3, 4.6] is corrected and generalized. We apply these results to G-transitive blocks of the universal enveloping algebras of restricted p-Lie algebras. Finally we deduce a condition for a smash product of a local basic algebra Λ with a commutative semi-simple group algebra to have components with Euclidean tree class, in terms of the components of the Auslander-Reiten quiver of Λ. In the last part we introduce and analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their Auslander-Reiten quiver is determined. We use these results to show that certain algebras are piecewise hereditary. Also a necessary condition for the existence of components of Euclidean tree class is deduced. We determine components that contain shift periodic complexes.

In this thesis, I apply methods from the representation theory of finite dimensional algebras to the study of versal and universal deformation rings. The main idea is that more sophisticated results from representation theory can be used to arrive at a deeper understanding of deformation rings. Such rings arise naturally in a variety of problems in number theory and group representation theory. This thesis has two parts. In the first part, Λ is an arbitrary finite dimensional algebra over a field k. If V is a finitely generated Λ-module, I prove that V has a versal deformation ring R(Λ, V ). Moreover, if Λ is self-injective and the stable endomorphism ring of V is isomorphic to k, then R(Λ, V ) is universal. If additionally A is a Frobenius algebra and Ω(Λ) denotes the syzygy operator over Λ, I show that the universal deformation rings of V and Ω(V) are isomorphic. In the second part, I analyze a particular finite dimensional Frobenius algebra Λ over an algebraically closed field k for which all the finitely generated indecomposable modules can be described combinatorially by using certain words in Λ. I use this description to visualize the indecomposable Λ-modules in the stable Auslander-Reiten quiver of Λ and determine all the components of this stable Auslander-Reiten quiver which contain Λ-modules whose endomorphism ring is isomorphic to k. Finally I determine the universal deformation rings of all the modules in these components whose stable endomorphism ring is isomorphic to k.

We investigate various relationships between categories of functors. The major examples are given by extending some duality to a larger structure, such as an adjunction or a recollement of abelian categories. We prove a theorem which provides a method of constructing recollements which uses 0-th derived functors. We will show that the hypotheses of this theorem are very commonly satisfied by giving many examples. In our most important example we show that the well-known Auslander-Gruson-Jensen equivalence extends to a recollement. We show that two recollements, both arising from different characterisations of purity, are strongly related to each other via a commutative diagram. This provides a structural explanation for the equivalence between two functorial characterisations of purity for modules. We show that the Auslander-Reiten formulas are a consequence of this commutative diagram. We define and characterise the contravariant functors which arise from a pp-pair. When working over an artin algebra, this provides a contravariant analogue of the well-known relationship between pp-pairs and covariant functors. We show that some of these results can be generalised to studying contravariant functors on locally finitely presented categories whose category of finitely presented objects is a dualising variety.

Fondée sur l’approche proposée par Philip Auslander (2004), cette thèse est centrée sur la construction du personnage artistique de la chanteuse française Marianne Oswald (1901-1985), la relation dialogique qu’elle entretient avec ses publics et sa présence à travers la mémoire et l’histoire culturelle. Aujourd’hui méconnus, son non-conformisme, son style vocal parlé-chanté, sa nature engagée et sa force polarisante inspirent une importante élite artistique à Paris et lui valent le titre d’artiste la plus discutée des années 1930 dans la presse. À travers les concepts d’authenticité et de féminités, tels que relevés dans la presse artistique, l’autobiographie de la chanteuse et son répertoire musical, cette recherche explore l’incidence des rapports de réciprocité qui s’exercent entre le personnage, le public et la mémoire sur la réception et la création d’une multiplicité de représentations du personnage artistique et suggère que Marianne Oswald a tenu un rôle transformationnel dans l’histoire musicale et culturelle française.

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