O objetivo dessa Dissertação é detalhar resultados conhecidos de Cohomologia em Topos de Grothendieck. Para isso, apresentamos a Álgebra Homológica em seu contexto mais geral, através de Categorias Abelianas, introduzindo as principais noções da área como funtores derivados e sequências espectrais. Desenvolvemos também o essencial da Teoria de Topos, explicando como um topos de Grothendieck surge como uma certa generalização dos feixes de conjuntos e fornecemos aspectos lógicos dos topos elementares. Focamos sobretudo nos Topos de Grothendieck pois a partir deles podemos construir categorias abelianas com suficientes injetivos, as quais são necessárias para expressar os grupos de cohomologia. / The final objective of this Dissertation is to detail known results of Cohomology in Grothendieck Topos. For this, we present Homological Algebra in its more general context, through Abelian Categories, introducing the main notions of the area as derived functors and spectral sequences. We also develop the basics of the Topos Theory, explaining how a Grothendieck Topos arises as a certain generalization of sheafs and we provide logical aspects of the elementary topos. We focus mainly in the Grothendieck Topos because from them we can construct abelians categories.

Sea R un anillo asociativo no unitario. Un módulo M se dice firme si es isomorfo de forma canónica al producto tensorial sobre R de R por M. La categoría formada por los módulos firmes es una generalización natural de la categoría de módulos unitarios para anillos unitarios.Una propiedad fundamental y que permanecía como problema abierto era la abelianidad de la categoría de módulos firmes. En la memoria se prueba que en general la categoría no es abeliana, mostrando un ejemplo de anillo asociativo R y de un monomorfismo que no es núcleo de ningún otro morfismo de la categoría. Se realiza un estudio profundo de la categoría de módulos firmes y de multitud de propiedades equivalentes a la abelianidad, así como otras propiedades más débiles y que tampoco se cumplen en general. / Let R a nonunital ring. A module M is set to be firm if it is isomorphic in the canonical way to the tensor product about R of R by M. The category of firm modules generalizes the usual category of unital modules for a unital ring.It was a open problem if the category of firm modules is an abelian category. We prove that, in general, this category is not abelian, and we find a ring and a monomorphism that is not a kernel in this category. The category of firm modules has been estudied in detail. We have deeply analyzed several properties equivalent to be abelian, and some others with weaker restrictions that are not satisfied in general

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.

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.