Balanced Pairs And Relative Singularity Categories

Derived categories play an important role in homological algebra, representation theory, algebraic geometry, mathematical physics and so on. The idea of derived cat-egories originated from Grothdieck around 1960. Later on, his student Verdier in-troduced the notion of triangulated categories and developed the localization theory of triangulated categories, which gave the concrete construction of derived categories. Derived categories behave better in dealing with some homological problems, for in-stance, one can regard derived functors as morphisms in derived categories, which bring a great convenience in computing and understanding and derived functors are well de-fined regardless of projective and injective objects. In this paper, we mainly study the following two aspects:balanced pairs and relative singularity categories, both of which have a close relation with derived categories.This paper is divided into four chapters.In Chapter 1, main results and preliminaries are stated.In Chapter 2, we introduce the notions of balanced pairs and for a given balanced pair [x,y) the notion of cotorsion pairs relative to(x,y) is given.Then we give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. At last, we prove that if the x-resolution dimension of y(resp.y-coresolution dimension of x) is finite,then the bounded homotopy category of y (resp.x) is contained in that of x(resp.y).In Chapter 3,we give the notion of the right l-singularity category of an a-belian category A relative to a subcategory L(?)A.In particular, when L(?)A is contravariantly finite and admissible, the right L-singularity category DRL-sg(A) is studied.We give a sufficient condition when this category is triangulated equivalent to the stable category of the Gorenstein category y(L) of L.In Chapter 4,we give a sufficient condition for the relative singular equivalences of categories being an invariant under relative derived equivalences.