Gorenstein-injective Derived Categories

Derived categories are introduced by Grothendieck in the sixties of last century. In the past decades, the theory and application of the derived cat-egories have been developed greatly, becoming a new research direction of algebra. On the other hand, homological algebra, especially Gorentein ho-mological algebra, has been developed to an advanced level. In homological algebra, projective and injective modules play a basic role. In Gorentein ho-mological algebra, they are replaced respectively by Gorenstein projective and Gorenstein injective modules, introduced by E.E. Enochs and O.M.G.Jenda in [1]. Note that the concepts have also been generalized to Abel category. It is natural to have a theory of the Gorenstein-injective derived category. In the theory of triangulated categories, we can obtain a new triangulated category by Verdier’s localization from a saturated multiplicative system of a triangulated category.One gets in the same way the derived category from the saturat-ed multiplicative system of quasi-isomorphisms of homotopy category. The Gorenstein-injective derived category is introduced and studied via Verdier’s localization of homotopy category respect to the saturated multiplicative sys-tem of (?)I quasi-isomorphisms. In this paper we study Gorenstein-injective derived categories, which intended to close a gap of the corresponding version of derived categories in Gorentein homological algebra.This thesis consists of three chapters.In chapter one, we recall some notions, backgrounds needed in the sequel, and finally, list the main results of this thesis.In chapter two, firstly, we give some well known results in triangulated categories and homotopy categories, by these facts we get some results which can be used in Abel category,and then we give a definition of Gorenstein-injective object. secondly, we give some characterizations and corresponding properties of Gorenstein-injective objects, finally, we obtain a result, that is the minimal (?)I resolution of every object has the same length. In chapter three, we give the definition of Gorenstein-injective derived category firstly. Then we introduce the relations between the Gorenstein-injective derived category and the usual derived category, and some analogs of the basic results in the usual derived category are proved in the Gorenstein-injective derived category. Secondly, as applications of the previous results, we explicitly describe the characterizations of the bounded Gorenstein-injective derived category on a Gorenstein ring and a finite-dimensional k—algebra. The corresponding bounded Gorenstein-injective derived category are realized as the homotopy categories of Gorenstein-injective objects. This interprets the Gorenstein derived functors as the Hom functor of the corresponding bounded Gorenstein-injective derived category.