Research On The Theory Of （Co）Tilting Comodules And The Related Problems

The representation theory of algebras is an important branch of algebra which emerged in the early 1970s,which mainly focuses on categories of modules over Artin algebra.And the main purpose of the representation theory of algebras is to characterize an algebra by the properties of the category of modules over the algebra.The main advantage of using module categories to study algebras is that the theory of categories and homological algebra can be applied,and the concept-s and theories of the abstract algebras can be visualized and concretized.The representation theory of associative algebras originated from that Hamilton used pairs of real numbers to describe complex numbers.In the 1830s,E.Noether first explained modules as representations,thus homological algebra and the theory of categories can be applied to the representation theory of algebras.Through the unremitting efforts of many generations of mathematicians at home and abroad,this theory has developed very quickly in recent years and gradually tend to per-fect.Since the comodule categories of finite-dimensional coalgebras is equivalent to the module categories of finite-dimensional algebras,the emphasis of represen-tation theory of coalgebras lies in the study of infinite dimensional coalgebras.Now,some achievements have been made in the study of the representation theo-ry of coalgebras.One of the open problems proposed by Simson is to develop the（co）tilting theory in coalgebras.Based on this,the specific research results are as follows:In chapter 2,we study the（pre）envelopes,finendo comodules,cotilting tor-sion classes and maximal cotilting comodules.Firstly,we give the definitions and properties of the（pre）envelopes and finendo comodules,and prove that the preen-velope torsion classes in comodule categories are consistent with the comodule classes generated by finendo comodules.Secondly,we introduce the definitions of cotilting comodules,classical（partial）cotilting comodules and study the relation-ship between them,then we introduce D-projective comodules,cotilting torsion classes and their properties,and give the Assem-Smal（?）theorem in comodule cat-egories.Finally,we introduce maximal cotilting comodules,envelope comodules,and prove that when the cotilting torsion class is an envelope class,it is uniquely represented by the envelope comodule.In chapter 3,we study the relationship between the precover torsion free classes and the comodules classeses cogenerated by cofinendo comodules,and the relationship between the tilting torsion free classes and the cover comodules in co-module categories.Firstly,we give the definitions and properties of（pre）covers and cofinendo comodules,and prove that the precover torsion free classes in comodule categories are consistent with the comodules classes cogenerated by cofinendo co-modules.Secondly,we introduce cotilting comodules over coalgebras,and prove that under the condition that C is a semi-perfect coalgebra,comodules class F is a tilting torsion free class if and only if F=Cogen（T）,where T is a faithful,cofinendo and F-injective comodule.Finally,we introduce maximal tilting co-modules,cover comodules,and prove that when the tilting torsion free class is a cover class,it is uniquely represented by the cover comodule.In chapter 4,we research the localization and colocalization of morphism-s,（pre）covers,tilting comodules,comodule classes Cogen_nM and torsion pairs in comodule categories.Firstly,we study the localization and colocalization of morphisms,and on this basis,we study the localization and colocalization of（pre）covers.Secondly,we research the localization and colocalization of tilting comodules,and prove that if Ce is a quasi-finite injective cogenerater,then M is a tilting eCe-comodule if and only if S（M）is a tilting C-comodule.In addi-tion,we study the localization and colocalization of comodule classes Cogen_nM,and prove that if Ce is a quasi-finite injective cogenerater,then eCe-comodule U（?）Cogen_∞M if and only if H（U）（?）Cogen_∞H（M）.Finally,we study the localization and colocalization of torsion pairs.