On Gorenstein Categories And Triangle-equivalences

Since1960s, Gorenstein homological algebra has been studied extensively and has become a vigorously active area of research. Enochs, Jenda and Garcia introduced the notations of Gorenstein projective modules (resp., Gorenstein injective modules) and Gorenstein projective complexes (resp., Gorenstein injective complexes), and in-vestigated the relationships between Gorenstein projective modules (resp., Gorenstein injective module) and Gorenstein projective complexes (resp., Gorenstein injective complex). Recently, the homotopy categories of projective modules and Gorenstein projective modules(resp., injective modules and Gorenstein injective modules), as tri-angulated categories, have been noticed by many authors. Iyengar and Krause in [30] gave the Iyengar-Krause’s equivalence, that is, for a commutative noetherian ring with a dualizing complex, the homotopy category of complexes of injective modules is equivalent to the homotopy category of complexes of projective modules. Subsequently, Chen in[13] gave a triangle-equivalence as an Iyengar-Krause’s equivalence between the homotopy category of complexes of Gorenstein projective modules and the homotopy category of complexes of Gorenstein injective modules. This dissertation is devoted to study homological properties of the Gorenstein projective objects and relative injective objects in abelian categories such as(?), Ch(?), etc. Subsequently, using cotorsion pairs as an important tool, we study triangle-equivalences between two homotopy categories of complexes.The thesis is divided into four chapters.In chapter1, we give some main results and preliminaries.In chapter2, we first define Gorenstein projective objects with respect to some subcategory of an abelian category(?) and corresponding subcategory of its chain com-plex category Ch(?), respectively. Then we study homological properties of them and investigate the relationships between the Gorenstein projective objects in (?) and the Gorenstein projective objects in Ch(?). Furthermore, under suitable conditions, we find a new model structure on Ch(?) by Hovey’s results in [29].In chapter3, we introduce the notions of relative injective objects in an abelian category(?) and in its chain complex category Ch(?), respectively. As an application, we extend the triangle-equivalence Kinj(R)→D(R), which was given in [34] in the triangulated category of R-modules, to the setting of any abelian category (?) with enough injective objects by the Verdier localization.A cotorsion pair in an abelian category (?) can induce two cotorsion pairs in the category of unbounded chain complexes Ch((?)). In chapter4, under some suitable conditions, we prove that a (co)localization sequence can be obtained by the induced cotorsion pairs. Further, we obtain some triangle-equivalences by the localization se-quence and the colocalization sequence. At last, we give some related applications with respect to the homotopy category of Gorenstein projective R-modules and the homotopy category of Gorenstein injective R-modules.