Teoria de Auslander-Reiten em categorias derivadas

• Autor(a) principal: Andrade, Aline Vilela
• Data de Publicação: 2014
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: LOCUS Repositório Institucional da UFV
• Texto Completo: http://locus.ufv.br/handle/123456789/4935
• topic: Modelos matemáticos; Teoria de Auslander-Reiten - Modelos matemáticos; Categoria derivada; Triângulo de Auslander-Reiten; Sequência de Auslander-Reiten; Mathematical models; Auslander-Reiten theory - Mathematical models; Derived category; Triangle Auslander-Reiten; Auslander-Reiten sequence; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::ALGEBRA

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.