?-Gorenstein Objects In Triangulated Categories

In 2018,Wei Jiaqun introduced the concept of ?-Gorenstein objects in a tri-angulated category T,where ? is a presilting subcategory of T,and proved that G? is closed under extensions and direct summands and finite direct sums.On the basis of Wei Jiaqun's work,this paper will further study the properties related to the subcategory G?.This paper consists of four chaptersIn chapter 1,we introduce the background and the main results of the thesis,and give some basic definitions and facts needed in the later chaptersIn chapter 2,we prove that any finite coresolution by objects in G? of an object gives rise to another finite coresolution such that one of the objects in the coresolu-tion belongs to G?,while all the rest objects are in add?.As an application of this result,we prove that any object K in G? admit two approximation triangles,where G? consists of all objects with a finite coresolution by objects of G?.In chapter 3,we further study adjoints between additive quotient categories related to the subcategory G?.More specificly,we construct an additive functor ?between additive quotient categories (?)?/[add?]and G?/[add?]and show that it is left adjoint to the inclusion functor G?/[add?]?(?)?/[add?].Similarly,we construct an additive functor ? between additive quotient categories (?)?/[G?]and (?)/[G?]and show that it is right adjoint to the inclusion functor (?)/[G?]?(?)?/[G?].In chapter 4,we construct a cellular tower with respect to the subcategory G?,and prove under certain conditions that such a cellular tower can be used to detect the value of G?-coresolution dimension.