Some Researches On Localization Of Categories

The notion of localizations was introduced by Grothedieck and Adams. Because of their fruitful algebric applications, the localizations have attracted many people's attention. This dissertation concentrates on the localizations, which includes three chapters altogether.The first chapter gives an introduction to the background and recent developmenton the localizations. Meanwhile, sum up the main results of this dissertation.The second chapter studies the quotient categories which are defined by the directed limt and the localizations which are given by the equivalent class of the roofs. We define a functor F from (?)/(?) to (?)[S^-1], and prove that two categories are equivalent.In the third chapter, some propositions are given, i.e, let C be an abelian category,W be a contravariantly finite subcategory of C and W (?)X(?)C. Then the additive category (C/W)/{X/W)≈C/X has a left triangulated structure.