Balanced Generalized Tilting Bimodules And Gorenstein Homological Dimension

With the development of the relative homological algebras in the past five decades, (generalized, co)tilting modules and Gorenstein homological algebras have been investigated widely by many authors and play important roles in the representation theory and the relative homology theory, respectively. We can bring together these by using the the nation of generalized G-dimension respect to a balanced bimodule, which was introduced by Auslander and Reiten. This thesis consists of the following parts:In chapter 1, the background, motivation and some notations are given.Chapter 2 deals with some homological finite properties of subcategories relative to a generalized tilting module, in which most of conclusions are well known. We obtain some equivalent characterizations of l.id(ω)< 1 and a cotilting modules of injective dimension zero by theω-tosionless(reflexive) properties of some subcategories, whereωis a balanced generalized tilting module.In the third chapter, we mainly investigate the k-torsionfree modules re-spect to a balance generalized tilting module and focus on the extension closure property of Tωk. Consequently, some classical results are extended. Further-more, we introduce a new dimension, called w-torsionfree dimension, and use it to obtain some equivalent characterizations of a generalized tilting module of finite injective dimension.In Chapter 4, the Gorenstein homological dimension of some special rings is investigated. We will establish the relation between the Gorenstein global dimension and the Gorenstein projective (injective) dimension of simple mod-ules. Finally, we generalize some classical results of homological algebras to relative case.It is worth pointing out that Avramov and Buchweitz proved a finite generated module over a noetherian ring is G-projective if and only if it has G-dimension zero. Hence, one can obtain some properties of a artinian ring by restricting the characterization of balanced generalized bimodules which is described by the generalized G-dimension of modules. we gain that the Goren-stein global dimension of an artinian ring can be controlled by the Gorenstein projective dimensions of all simple modules use the result above.which gener-ated the result in classical homological algebra.