Strongly ξ-gorenstein Projective Objects In Extriangulated Categories

Let the additive category C be an extriangulated category with IE-triangles proper class ξ,and it has enough ξ-projectives and enough ξ-injectives.This thesis mainly studies the homological properties of objects in extriangulated categories.The whole thesis is divided into two parts.Firstly,we discuss the basic homological properties of ξ-projective resolution of objects in extriangulated categories,and then proves that A→B→C→is an IE-triangle of C(-,P(ξ))-exact in ξ if A and B have C(-,P(ξ))-exact ξ-projective resolution,we can get a C(-,P(ξ))-exact ξ-projective resolution of C.Secondly,we introduce strongly ξ-Gorenstein projective objects and ξxt-stronglyξ-Gorenstein projective objects in extriangulated categories.And we study some basic homological properties of strongly ξ-Gorenstein projective objects and ξxt-strongly ξ-Gorenstein projective objects,and give the equivalent characterizations of strongly ξ-Gorenstein projective objects under extensions closed.We discuss the relations among the class of ξxt-strongly ξ-Gorenstein projective objects,stronglyξ-Gorenstein projective objects and ξ-Gorenstein projective objects.In particular,we proves that each ξ-Gorenstein pojctive object is the direct summand of a strong-ly ξ-Gorenstein projective object under the condition that extriangulated categories have infinite direct sums and IE-triangles proper class ξ is closed under direct sums.