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.

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.