Resolving Subcategories And The Recollements Of Abelian Categories

Resolving subcategories are important research objects in relative homological algebra.It is closely related to the contravariantly finite subcategories,thick subcat-egories,tilting theory and approximation theory.This notion has been introduced by Auslander and Bridger to prove that the category of totally reflexive modules is a re-solving subcategory of the category--R?-mod of finitely generated modules over a noethe-rian ring R.One of the main contents of this thesis is to give the finiteness criteria of the resolution dimension of resolving subcategories by studying the the properties of resolving subcategories and the resolving resolution dimension.Recollements have been introduced by Beilinson,Bernstein and Deligne first in the context of triangulat-ed categories.A well-known example of a recollement situation of abelian categories appeared in the construction of perverse sheaves given by MacPherson and Vilonen.Recollements of abelian categories provide a very useful framework for investigating the homological connections among these categories.We bring the resolving subcategories of the corresponding categories involved in recollements of abelian categories to inves-tigate the relationship of the resolving resolution dimension of abelian categories and apply to the Gorenstein homological algebra.This thesis is divided into five chapters.In Chapter 1,we provide the research background,motivation and main results.In Chapter 2,we recall some basic concepts related to this thesis.In Chapter 3,we study the finiteness criteria of the resolving resolution dimen-sion of resolving subcategories.Let A be an abelian category with enough projective objects,X a resolving subcategory of A and H an Ext-injective cogenerator for X.We study the properties of the subcategory consisting of all the objects of finite X-resolution dimension.Let M be an object of finite X-resolution dimension,we study the H-resolution dimension of M and discuss the relationship between H-resolution dimension and X-resolution dimension of M.As applications,we give the finiteness criteria of the homological dimensions in Gorenstein categories of modules and com-plexes.In Chapter 4,we investigate the relationship of the resolving resolution dimension of the corresponding abelian categories involved in a recollement of abelian categories.In particular,we give the relationship between the Gorenstein global dimension of the categories involved in a recollement of abelian categories.Moreover,in terms of the recollement of abelian categories(A,B,L),we study two invariants spli*and silp*(we denote by*the abelian categories A,B or L),which are closely related to the Gorenstein projective and Gorenstein injective dimensions of the abelian categorie*,where silp*is defined as the supremum of the injective lengths of projective objects of*,whereas spli*is the supremum of the projective lengths of injective objects of*.At last,we apply the above results to discuss the relationship of Gorensteinness of the categories involved in a recollement of abelian categories.In Chapter 5,we study some rings and triangular matrix artin algebras.Let R be a ring and e an idempotent element of R.We study the Gorenstein,quasi-Frobeniusand Gorenstein hereditary properties of the ring R and eRe.Let Λ=(0 B A AMB)be a triangular matrix artin algebra.We characterize the Gorenstein global dimensions of Λ in terms of the Gorenstein global dimensions of A and B.In particular,we get the Gorensteinness of A by the Gorensteinness of A and B.