The Theory Of Covers And Envelopes In Relative Homological Algebra

Approximation theory is the main part of relative homological algebra and repre-sentation theory of algebras,and its starting point is to approximate arbitrary objects by a class of suitable sub categories.Approximation theory goes back to the discov-ery of the injective hull by Reinhold Baer in 1940.Since the late 1950s,injective envelopes,projective covers as well as pure-injective envelopes,have successfully been used in module theory of arbitrary rings.Independent research by Auslander and Reit-en in the finite-dimensional case,and by Enochs and Xu for arbitrary modules,created a general theory of preenvelopes and precovers(or left and right approximations)of modules.In this paper,we mainly study the following topics:special precovered cat-egories of Gorenstein categories,cover and preenvelope by relative FP-injective and FP-flat complexes and the corresponding model structures,special precovering and special preenveloping ideals in extriangulated categories.This paper is divided into four chapters.In Chapter 1,main results and preliminaries are stated.In Chapter 2,we study the structure of the category which is special precovered by Gorenstein category.Let(?)be an abelian category and g(?))the subcategory of?)consisting of projective objects.Let l be a full,additive and self-orthogonal sub-category of?)with g(?))a generator,and let g(l)be the Gorenstein subcategory of?)relative to l.Then the right 1-orthogonal category g(l)?1 of g(l)is both projectively resolving and injectively coresolving in?).We also get that the subcate-gory SPC(g{l))of?)consisting of objects admitting special g(l)-precovers is closed under extensions and l-stable direct summands(*).Furthermore,if l is a generator for g(l)?1,then we have that SPC(g(l))is the minimal subcategory of(?)containing g(l)?1 ? g(l)with respect to the property(*),and that SPC(g(l))is l-resolving in?)with a l-proper generator l.In Chapter 3,we study homological and homotopical aspects of injective and flat chain complexes relative to chain complexes of type FPn.We introduce the notions of FPn-injective and FPn-flat complexes,and describe homological dimensions associated to them.We show that for these dimensions there exists a characterization analogous to that for the injective and flat dimensions of complexes.Taking advantage of the relation between the FPn-injective and FPn-flat dimensions via Pontrjagin duality,and using the theory of cotorsion,pairs,we construct(pre)covers and(pre)envelopes associated to these dimensions,establishing thus an approximation theory relative to complexes of finite type.We also construct some model structures on complexes from modules with bounded FPn-injective and FPn-flat dimensions,and analyze several situations in which it is possible to connect these structures via Quillen functors.In Chapter 4,we introduce and study relative phantom morphisms in extriangulat-ed categories.Then using their properties,we show that if(C,E,s)is an extriangulated category with enough injective objects and projective objects,then there exists a bijec-tive correspondence between any two of the following classes:(1)special precovering ideals of C;(2)special preenveloping ideals of C;(3)additive subfunctors of E having enough special injective morphisms;and(4)additive subfunctors of E having enough special projective morphisms.Moreover,we show that if(C,E,s)is an extriangulated category with enough injective objects and projective morphisms,then there exists a bijective correspondence between the following two classes:(1)all object-special pre-covering ideals of C;(2)all additive subfunctors of IE having enough special injective objects.