Gorenstein Modules And Hopf Actions

Enochs, Jenda and their coauthors introduced and studied Gorenstein projective, injective and flat modules and developed relative homological algebra. The aim of this thesis is to study further Gorenstein modules and to investigate some algebra structures in Hopf algebra by homological methods.In Chapter 1, some symbols and preliminaries are given.In Chapter 2, we introduce (x,y)-Gorenstein projective and injective modules which are a generalization of Enochs' Gorenstein projective and injective modules re-spectively. We mainly discuss the relations among various (x,y)-Gorenstein projective and injective modules.In Chapter 3, we introduce (?)p-projective and (?)I-injective modules and give some characterizations. Our main aim is to discuss the "global" dimension GPID(R) sup{id(M)|M(?)RM with M Gorenstein projective}.In Chapter 4, we introduce C-free modules and Gorenstein C-flat modules, the latter is different from C-Gorenstein flat modules [30], where C is a semidualizing module. The aim of this chapter is to discuss C-free, C-flat, C-Gorenstein flat and Gorenstein C-flat modules and their relations.In Chapter 5, we discuss the homological dimension and Gorenstein dimension of L-R smash products A(?)H, where H is a Hopf algebra and A is an H-bimodule algebra. Several well-known results are generalized. Moreover, we explore the stability of Gorenstein projective (flat) precovers and Gorenstein injective preenvelopes between the category of left A-modules and the category of left A(?)H-modules.