On Gorenstein Properties Of Comodules And Modules

In classical homological algebra, projective, injective and flat dimensions of mod-ules are important and fundamental research objects. As a generalization of the notion of projective dimension of modules, Auslander and Bridger [3] introduced in 1969 the G-dimension, G-dimM, for every finitely generated R-module M (see also [2]) for a two-sided Noetherian ring R. Several decades later, Enochs, Jenda, and Torrecillas [20,21,23] extended the idea of Auslander and Bridger. They defined Gorenstein pro-jective, injective and flat modules, and the related Gorenstein homological dimension for any module M. Avramov, Buchweitz, Martsinkovsky and Reiten proved that a finitely generated module over a Noetherian ring is Gorenstein projective if and only if G-dimRM= 0 [10].In the recent years, these Gorenstein modules have become a vigorously active area of research. In particular, Asensio, Enochs et al. ([1,24]) defined Gorenstein injective and projective C-comodules in the category of comodules. Along the same lines, Gorenstein injective and projective dimensions of a comodule were also introduced and studied. Let X be a class of right R-modules that contains all projective right R-modules, Bennis et al.([7]) introduced the notion of X-Gorenstein projective right R-modules.In this dissertation, we introduce and study Gorenstein coflat comodules and weakly Gorenstein injective (resp. coflat) comodules over any coalgebra C. We also introduce and study y-Gorenstein injective right R-modules and y-Gorenstein flat left R-modules, where y is a class of right R-modules that contains all injective right R-modules. Finally we study the relationship of Gorenstein flat (cotorsion) dimen-sions, FP-injective (FP-projective) dimensions and cotorsion pairs between A-Mod and A#H-Mod. This paper consists of four chapters.In Chapter 1, some backgrounds and preliminaries are given.Chapter 2 is devoted to introducing Gorenstein coflat comodules and weakly Gorenstein injective (resp. coflat) comodules over any coalgebra C. We prove that, for a left semiperfect coalgebra C, a left C-comodule M is (weakly) Gorenstein coflat if and only if M is (weakly) Gorenstein injective. We also study the existence of pre-covers and preenvelopes by weakly Gorenstein injective comodules. It is shown that every left C-comodule has a weakly Gorenstein injective preenvelope (cover) over a left semiperfect coalgebra C.Chapter 3 is devoted to studying y-Gorenstein injective right R-modules and y-Gorenstein flat left R-modules, where y is a class of right R-modules that contains all injective right R-modules. We show that principal results on Gorenstein projective, injective and flat modules remain true for X-Gorenstein projective right R-modules, y-Gorenstein injective right R-modules and y-Gorenstein flat left R-modules..Let H be a finite-dimensional Hopf algebra over a field k and A a left H-module algebra. In Chapter 4, we prove that over a right coherent ring A (resp. any ring A), if A#H/A is separable andφ:A→A#H is a splitting monomorphism of (A, A)-bimodules, l.Gwd(A)= l.Gwd(A#H) (resp. l.cotD(A)=l.cotD(A#H) and l.Gcd(A)= l.Gcd(A#H)). We also study the FP-injective (FP-projective) dimen-sions under actions of finite-dimensional Hopf algebras. We get that if A#H/A is separable andφ:A→A#H is a splitting monomorphism of (A,A)-bimodules, then l.FP-dim(A)=l.FP-dim(A#H) and l f pD(A)=lfpD(A#H). Finally, we give a con-nection of cotorsion pairs between A-Mod and A#H-Mod.