Studies On Torsion Pairs And Quotient Categories

This thesis for Ph.D degree is on triangulated quotient categories and their applications. It mainly consists of the following three parts.1. For any triangulated categories and a functorially finite subcategory ω, J(?)rgensen showed that the quotient category C/ω is a pretriangulated category. When τω=ω, where τ is the Auslander-Reiten translation functor, the quotient category C/ω is a triangulated category. In Chapter 3, we study torsion pair in the above pretriangulated category and triangulated category, and obtain necessary and sufficient conditions for torsion pair in a pretriangulated category and triangulated category.2. Iyama and Yoshino introduced the notion of D-mutation pairs in tri-angulated categories. It is mainly used to solve some about Cohen-Macaulay modules problems. After the concept is given, the theory research became a hot spot. Based on this, In Chapter 4, The notion of D-mutation pairs of subcate-gories in a triangulated category is defined in this article, but we don’t assume that D is rigid here. We generalize Iyama and Yoshino result, i.e., If (Z, Z) is a D-mutation pair in a triangulated category C and Z is extension closed, then the quotient category E/D carries naturally a triangulated structure. At the same time, if D and Z satisfy certain conditions in C with a Serre functor, then the quotient category Z/D has a Serre functor, In particular, C is an n-Calabi-Yau, then the quotient category Z/D is also an n-Calabi-Yau. Moveover, We study mutations of n-cluster-tilting subcategories and get the following result: If (X, y) is D-mutation pair, then X is n-cluster-til ting subcategories if and only if y is n-cluster-tilting subcategories; We give a one-one correspondence between cotorsion pairs in C and cotorsion pairs in the quotient category Z/D.3. In Chapter 5, we study the relationship of torsion pairs among three triangulated categories in a recollement. We show that if a triangulated category V admits a recollement relative to triangulated categories D’ and D", we can get torsion pairs in D from torsion pairs in D’ and D". On the contrary, we show that certain torsion pairs in D can induce torsion pairs in D’ and D".