Some Research On Pullback And Localization Of Categories

The notion of pullback in ring theory and in representation theory of algebras is quite basic and has been studied by many authors. And the notion of localization of abelian categories by serre classes was introduced by Gabriel and developed by Grothendieck, Verdier, MiyachiThe first chapter gives an introduction to the background and recent developmenton the pullback and localization of abelian categories. Meanwhile, sum up the main results of this dissertation.In the second chapter, given two homomorphisms of rings j₁:R₁→R′and j₂:R₂→R′,a new ring R called the pullback of R₁ and R₂ over R' is constructed. Let T denote the category whose objects are the triples (M₁, M₂,α) where M_i∈R_i-Mod, i=1, 2, andαis an R′-isomophism. The funtors P : T→R-Mod and S:R-Mod→T are proved to be an adjoint pair and have quasi-inverse relation in the special case of the rings of residue classes of modulo integers.In the third chapter, we will consider the localization of categories of modules over pullbacks. In section 2, we show that the full subcategory T_R consisting of pullback modules in R-Mod is a Serre class, and then we give the corresponding Serre classes T_R1,T_R2 and T_R′ via Tr. In section 3, we study the localization of categories of modules by serre classes and investigate pullback properties of functors.