Deformation theory studies the behavior of mathematical objects, such as representations or modules, under small perturbations. This theory is useful in both pure and applied mathematics and has been used in the proof of many long-standing problems. In particular, in number theory Wiles and Taylor used universal deformation rings of Galois representations in the proof of Fermat's Last Theorem. The main motivation for determining universal deformation rings of modules for finite dimensional algebras is that deep results from representation theory can be used to arrive at a better understanding of deformation rings. In this thesis, I study the universal deformation rings of certain modules for algebras of dihedral type of polynomial growth which have been completely classied by Erdmann and Skowronski using quivers and relations. More precisely, let κ be an algebraically closed field and let λ be a κ-algebra of dihedral type which is of polynomial growth. In this thesis, first classify all λ-modules whose stable endomorphism ring is isomorphic to κ and which are given combinatorially by strings, and then I determine the universal deformation ring of each of these modules.

Torsion pairs were introduced by Dickson in 1966 as a generalization of the concept of torsion abelian group to arbitrary abelian categories. Using torsion pairs, we can divide complex abelian categories in smaller parts which are easier to understand. In this thesis we discuss torsion pairs in the category of modules over a finite-dimensional algebra, in particular we explore the relation between torsion pairs in the category of all modules and torsion pairs in the category of finite-dimensional modules. In the second chapter of the thesis, we present the analogue of a classical theorem of Auslander in the context of τ-tilting theory: for a finite-dimensional algebra the number of torsion pairs in the category of finite-dimensional modules is finite if and only if every brick over such algebra is finite- dimensional. In the third chapter, we revisit the Ingalls-Thomas correspondences between torsion pairs and wide subcategories in the context of large torsion pairs. We provide a nice description of the resulting wide subcategories and show that all such subcategories are coreflective. In the final chapter, we describe mutation of cosilting modules in terms of an operation on the Ziegler spectrum of the algebra.

Orientador: Antonio Jose Engler / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T18:56:31Z (GMT). No. of bitstreams: 1 Ferreira_MauriciodeAraujo_M.pdf: 1033477 bytes, checksum: 8d697b5cdeb1a633c1270a5e2f919de7 (MD5) Previous issue date: 2006 / Resumo: Neste trabalho, estudamos as álgebras biquaterniônicas, que são um tipo especial de álgebra central simples de dimensão 16, obtida como produto tensorial de duas álgebras de quatérnios. A teoria de formas quadráticas é aplicada para estudarmos critérios de decisão sobre quando uma álgebra biquaterniônica é de divisão e quando duas destas álgebras são isomorfas. Além disso, utilizamos o u-invariante do corpo para discutirmos a existência de álgebras biquaterniônicas de divisão sobre o corpo. Provamos também um resultado atribuído a A. A. Albert, que estabelece critérios para decidir quando uma álgebra central simples de dimensão 16 é de fato uma álgebra biquaterniônica, através do estudo de involuções. Ao longo do trabalho, construímos vários exemplos concretos de álgebras biquaterniônicas satisfazendo propriedades importantes / Mestrado / Algebra / Mestre em Matemática