Resolutions And Strongly ?-gorenstein Projective Objects In A Triangulated Category

We introduce and study several main properties and generalizations of ?-Gorenstein projective objects in a triangulated category.In the first part, we first define strongly ?-exact complex and strongly ?-projective resolution, we provide a method to construct a resolution of the first or third term in a triangle. Furthermore, we get the stability of ?-Gorenstein projec-tive objects in a new method.In the second part, we study the typical properties of strongly ?-Gorenstein projective objects based on the definition of the strongly ?-Gorenstein projec-tive objects, we get that the class of ?-Gorenstein objects is closed under exten-sion and the class of strongly ?-Gorenstein projective objects is not closed un-der extension. Then we introduce the class between ?-Gorenstein objects and strongly ?-Gorenstein objects, which is called Ext-SGP(?), based on the relations?-projective objectsCstrongly ?-Gorenstein projective objects(?)?-Gorenstein pro-jective objects and we give their properties.