This thesis is divided into two parts. The rst part studies homological algebra of quasihereditary algebras, with the underlying theme being to understand properties of the Yoneda algebra of standard modules. We will rst show how homological properties of a quasi-hereditary algebra are carried over to its tensor products and wreath products. We then determine the extgroups between indecomposable standard modules of a Cubist algebra of Chuang and Turner. We will also determine generators, hence the quiver, of the Yoneda algebra of standard modules for the rhombal algebras of Peach. We also obtain a higher multiplication vanishing condition for certain rhombal algebras. The second part of this thesis studies the notion of simple-minded systems, introduced by Koenig and Liu. Such systems were designed to generate the stable module categories of artinian algebras by extension, in the same way as the sets of simple modules. We classify simple-minded systems for representation- nite self-injective algebras, and establish connections of them to various notions in combinatorics and related derived categories. We also look at the notion of simple-minded systems de ned on triangulated categories, and obtain some classi cation results using a connection between the simple-minded systems of a triangulated category and of its orbit category.

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.

This thesis consists of three papers about N-complexes and their uses in categorification. N-complexes are generalizations of chain complexes having a differential d satisfying dN = 0 rather than d2 = 0. Categorification is the process of finding a higher category analog of a given mathematical structure. Paper I: We study a set of homology functors indexed by positive integers a and b and their corresponding derived categories. We show that there is an optimal subcategory in the domain of every functor given by N-complexes with N = a + b. Paper II: In this paper we show that the lax nerve of the category of chain complexes is pointwise adjoint equivalent to the décalage of the simplicial category of N-complexes. This reveals additional simplicial structure on the lax nerve of the category of chain complexes which provides a categorfication of the triangulated homotopy category of chain complexes. We study this in general and present evidence that the axioms of triangulated categories have a simplicial origin. Paper III: Let n be a product of two distinct prime numbers. We construct a triangulated monoidal category having a Grothendieck ring isomorphic to the ring of n:th cyclotomic integers.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The aim of this dissertation is to present a simpli cation of the proof of the following result obtained rst by Happel in [3]: If A is a nite-dimensional algebra over a eld algebraically closed K, then there is a triangulated, full and faithful functor of triangulated categories H : Db(modA) ! modA^, where A^ is the repetitive algebra obtained from A, which is also dense if A is of nite global dimension. We begin with a succinct presentation of the categorical language, approaching in general terms on the localization of categories, triangulated categories and their localizations, and nally derived categories, which are localized and triangulated categories. We also introduce the stable category of modules of a repetitive algebra A^. In the last chapter, we demonstrate the main result with the help of a result found in [8], in addition to the previously mentioned concepts. / O objetivo dessa disserta c~ao e trazer uma simpli ca c~ao da demonstra c~ao do seguinte resultado obtido primeiramente por Happel [3]: Se A e uma K- algebra de dimens~ao nita, ent~ao existe um funtor pleno, el e triangulado H : Db(modA) ! modA^, onde A^ e a a lgebra repetitiva obtida de A, que e tamb em denso se A e de dimensa~o global nita. Iniciamos com uma apresenta c~ao sucinta da linguagem categ orica, abordando de maneira geral sobre localiza c~ao de categorias, categorias trianguladas e suas localiza c~oes, e nalmente categorias derivadas, que s~ao categorias localizadas e trianguladas. Tamb em introduzimos a categoria est avel de m odulos da algebra repetitiva de A. No ultimo cap tulo, demonstramos o resultado principal com o aux lio de um resultado encontrado em [8], al em dos conceitos citados anteriormente. / São Cristóvão, SE

The Grothendieck group is an interesting invariant of an exact category. It induces a decategorication from the category of essentially small exact categories (whose morphisms are exact functors) to the category of abelian groups. Similarly, the triangulated Grothendieck group induces a decategorication from the category of essentially small triangulated categories (whose morphisms are triangulated functors) to the category of abelian groups. In the case of an essentially small abelian category, its Grothendieck group and the triangulated Grothendieck group of its bounded derived category are isomorphic as groups via a natural map. Because of this, homological algebra and derived functors become useful in surprising ways. This thesis is an expository work that provides an overview of the theory of Grothendieck groups with respect to these decategorications.

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