Some Studies On Triangulated Categories

The notion of triangulated categories and derived categories was induced by A.Grothendieck and J.L.Verdier in1960s. ver the past decades, triangulat-ed categories have made their way into many different parts of mathematics and have become indispensable in many different areas of mathematics. This article belongs to the domain in the theory of triangulated categories in the representation theory of algebras. The main purposes of this thesis is to study the properties of triangulated categories by using the methods of homological algebra and representation theory.This thesis consists of four chapters.In chapter one, we recall some notions and classic results, which are needed in the sequel, in the theory of triangulated categories.In chapter two, we study the subfactor categories of triangulated cate-gories and get the follow main results:●Let T be a triangulated category, X ()? A subcategories of T. If X is a covariantly finite sub category of A and any X-monomorphism of A has a cone in A, then the subfactor category A/[X] admits a right triangulated structure.●Give a characterization of D-mutation pair in triangulated categories by applying the theory of subfactor categories.In chapter three, we prove the conjecture Ⅱ.1.9of [15], which said that any maximal rigid object without loops or2-cycles in its quiver is a cluster tilting object in a connected Hom-finite triangulated2-CY category T, to be true.In chapter four, we study in triangulated categories the theory of mor-phisms determined by objects which is introduced by M.Auslander in the cat-egory of modules. In the paper [45] H.Krause proved that for a Hom-finite essentially small and Krull-Schmidt R-linear triangulated category C with a Serre functor S every morphism f is right determined by an object in C (The-orem4.2,[45]), where R denotes a commutative artinian ring. In this chapter we give the form of the minimal determiner D(f) for any morphism f in a triangulated category C with a Serre functor S, which is a complement to the Theorem4.2of H.Krause.