Torsion Pairs And N-cluster Tilting Categories In Triangulated Categories

This paper focuses on the properties and correlations torsion pairs, cotor-sion pairs, and n-cluster tilting subcategories. It consists of four chapters.In first chapter, we recall some basic definitions and background knowl-edge to understand the sequel. Meanwhile, the main conclusion of this thesis was given.In second chapter, firstly, we introduced the notion and properties of tor-sion pairs in triangulated category. Based on these notion and properties, we derived some conclusions on covariantly(contravariantly) finite and extension closed subcategory and torsion pairs. Secondly, we introduce the definitions of cotorsion pairs and deduced the properties and equivalent conditions.In third chapter, we introduced the definitions of n-rigid and n-cluster tilting in triangulated categories. Based on these definitions, some conclusion on triangulated category were drawn. With these conclusions, we derived the equivalent conditions between torsion pairs and cluster tilting subcategories. Secondly, we introduced the notions and properties of D-monic and D-epic, we proved the correlations of objective and morphism.In forth chapter, we introduced the definitions of Serre function, n-rigid torsion pairs, and core in triangulated categories. With these properties, we derived some properties and correlations among torsion pairs (cotorsion pairs), core and rigid subcategories.