Gorenstein Objects In Extriangulated Categories

This thesis mainly studies the relative Gorenstein objects in the extriangulat-ed categorywith a proper classand the related properties of these objects.In the first part,we define the notion of the?-Gorenstein projective reso-lution(see Definition 3.19),and study the relation between-projective resolu-tion and?-Gorenstein projective resolution for any objectin(see Theorem3.21),i.e.has a(-,())-exact-projective resolution if and only ifhas a(-,())-exact?-Gorenstein projective resolution.In the second part,we define a particular?-Gorenstein projective object inwhich called-9)-strongly Gorenstein projective object(see Definition 4.1).On this basis,we study the relation between-8)-strongly Gorenstein projective ob-jects and-9)-strongly Gorenstein projective objects whenever8)=9)(see Theo-rem 4.6),and give some equivalent characterizations of-9)-strongly Gorenstein projective objects(see Theorem 4.8).In the third part,we introduce the notion of the proper?-Gorenstein projec-tive resolution(see Definition 5.5),then the Horseshoe Lemma and Comparison Theorem of the proper?-Gorenstein projective resolition version are given.On the foundation of this,we define the?-Gorenstein derived functorxt⁴)(-,-)on certain conditons(see Definition 5.20),and give some equivalent character-izations for the objects with finite?-Gorenstein projective dimension and finite?-Gorenstein injective dimension by usingxt⁴)(-,-)(see Proposition 5.27 and5.28).