On Finite Resolutions In Extriangulated Categories

In this thesis,we study mainly finite resolutions and via additive quotient categories in extriangulated categories.In chapter 1,we introduce the background and the main results of the thesis,and give some basic definitions and facts needed in the later chapters.In chapter 2,we introduce the concept of a weak-cogenerator(weak-generator)in extriangulated categories.We discuss the approximation theory and investigate finite resolutions in extriangulated category.More specifically,let(C,E,s)be an extriangulated category.Suppose that X and W are two subcategories in C.We show that if W is both a weak-cogenerator and a weak-generator in X,then any finite resolution by objects in X of an object gives rise to another finite resolution such that one of the objects in the resolution belongs to X,while all the rest objects belong to W.In chapter 3,we investigate finite resolutions in extriangulated categories via additive quotient categories.We construct an additive functor ? between additive quotient categories X/[W]and X/[W]and show that it is right adjoint to inclusion functor X/[W]?X/[X].Similary,we construct an additive functor ? between additive quotient categories X/[X]and W/[X]and show that it is left adjoint to inclusion functor W/[X]? X/[X].