Gorenstein Object In Triangulated Categories

As an important branch of the representation theory of algebras, triangulated cate-gories are the bridge between representation theory of algebras and algebraic geometry.There are widely used in many areas of mathematics and Physics. This thesis for Masterdegree is to study Gorenstein projective object in triangulated categories. The main resultsobtained are as follows.In Chapter3, some equivalent new descriptions are given forξ-Gorenstein projectivedimension of objects, based on theξ-Gorenstein derived functor, specialξ-exact complexand Schanuel class relative to GP(ξ) in triangulated categories. We give a equivalent condi-tion for anξ-n-Gorenstein triangulated category.In Chapter4, we introduce a particular case ofξ-Gorenstein projective objects intriangulated categories, that is, n-stronglyξ-Gorenstein projective objects for any integern≥1. We show the relation between n-stronglyξ-Gorenstein projective objects and m-stronglyξ-Gorenstein projective objects in the case n m. We also give an condition for ann-stronglyξ-Gorenstein projective object isξ-projective.In Chapter5, we introduce a particular case of objects in triangulated categories,that is,(n,m)-stronglyξ-Gorenstein projective objects ((n,m)-ξ-SG-projective objects forshort), and we study their syzygies. We prove that an object X hasξ-Gorenstein projectivedimension at most m if and only if X⊕G is (1,m)-ξ-SG-projective for some Gorensteinprojective object G.