【作者】 孟凡云；

【导师】 丁南庆；

【作者基本信息】 南京大学 ， 基础数学， 2011， 博士

【摘要】 在经典同调代数中,模的投射维数、内射维数和平坦维数是重要且基本的研究对象。作为模的投射维数的概念的推广,Auslander和Bridger于1969年在文献[3]中对双侧Noether环R上的有限生成模定义了G-维数的概念(又见文献[2])。几十年后,Enochs,Jenda和Torrecillas [20,21,23]推广了Auslander和Bridger的思想。他们定义了Gorenstein投射,内射和平坦模,并引入了任意模的Gorenstein同调维数的概念。Avramov, Buchweitz, Martsinkovsky和Reiten证明了,对Noether环上的有限生成模M,M是Gorenstein投射的当且仅当它的Gorenstein投射维数是0[10]。近年来,这些Gorenstein模已经有了很多形式的推广。特别地,Asensio, Enochs等人([1,24])在余模范畴中定义了Gorenstein内射和投射余模,并且研究了余模的Gorenstein内射和投射维数。而Bennis等人([7])引入了X-Gorenstein投射模的概念,其中X是包含投射模的模类。本文中,我们在任意的余代数C上定义并研究了Gorenstein余平坦和弱Gorenstein内射(余平坦)余模。同时,我们引入并研究了y-Gorenstein内射右R-模和y-Gorenstein平坦左R-模,其中y是包含所有内射右R-模的右R-模类。最后,我们研究了Gorenstein平坦(余挠)维数,FP-内射(FP-投射)维数和余挠对在A-Mod和A#H-Mod之间的关系。全文共分四章,内容如下第一章是引言,主要介绍了问题的背景和预备知识。在第二章中,我们在任意的余代数C上定义了Gorenstein余平坦和弱Gorenstein内射(余平坦)余模,并且证明了,在左半完全余代数C上,左C余模M是(弱)Gorenstein余平坦余模当且仅当M是(弱)Gorenstein内射余模。同时,我们研究了弱Gorenstein内射余模预覆盖和预包络的存在性问题。我们证明了在左半完全余代数C上,任意的C余模都有一个弱Gorenstein内射预包络(覆盖)。第三章我们引入了y-Gorenstein内射右R-模和y-Gorenstein平坦左R-模,其中y是包含所有内射右R-模的右R-模类。我们证明了关于Gorenstein投射,内射和平坦模的主要结果对于X-Gorenstein投射右R-模,y-Gorenstein内射右R-模和y-Gorenstein平坦左R-模仍然成立。设H是域k上的有限维Hopf代数,A是左H-模代数。在第四章中,我们证明了在右凝聚环A(或任意环A)上,如果A#H/A可分并且φ:A→A#H是可裂的(A,A)双模单同态,那么l.Gwd(A)=l.Gwd(A#H)(或l.cotD(A)=l.cotD(A#H)和l.Gcd(A)= l.Gcd(A#H))。同时,我们研究了FP-内射(FP-投射)维数在有限维Hopf代数下的作用。我们证明了在任意环A上,如果A#H/A可分并且φ：A→A#H是可裂的(A,A)双模单同态,那么l.FP-dim(A)=l.FP-dim(A#H),并且l f p D(A)=l fpD(A#H).最后,我们给出了余挠对在A-Mod和A#H-Mod之间的关系。更多还原

【Abstract】 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.更多还原