Gorenstein Projective Covers, Preenvelopes And Relative Cohomology

The notions of (pre)covering class and (pre)enveloping class of modules play crucial roles in relative homological algebra. These classes can be viewed as a generalization of those of projective and injective modules, and they allow us to construct various proper resolutions which are different to projective or injective resolutions. Around 2000, by showing the existence of plenty of such classes of modules, Enochs and Jenda et al. reinforced the foundation of relative homological algebra, which was originated from the work of Eilenberg and Moore. Using the corresponding proper resolutions, we obtain more general concepts of dimensions, derived functors, (co)homology modules and balanced functors. This enriches the classical homological theory.This thesis is divided into four chapters.Chapter 1 contains an introduction and a description of main results, the notation that will be needed in the sequel is introduced, too.Chapter 2 is dedicated to investigating the conditions for the existence of Goren-stein projective covers and preenvelopes. We show that it shares many similarities with the case in classical ring theory:in fact, for the rings over which a module is Gorenstein flat if and only if it is Gorenstein projective, the class of Gorenstein projective modules is covering. Furthermore, we characterize the rings over which the class of Gorenstein projective modules is preenveloping. The existence of some other Gorenstein modules precovers and preenvelopes is also studied.In Chapter 3, we compare relative cohomology theories arose from using different proper resolutions of modules. Criteria for the vanishing of such distinctions is given in certain cases, and we show that this is related to the generalized Tate cohomology theory. We also demonstrate that the two balance properties occurring in the two different cohomology theories are actually equivalent in some cases. As applications, we recover many results obtained earlier in various contexts. At last we investigate derived functors with respect to the Auslander class and Bass class.In Chapter 4, we study the dimension and precovering theories of χ-Gorenstein projective modules for a certain class χ of R-modules.