Some Topics Of Gorenstein Homological Theories

Inspired by Auslander's definition of G-dimension[6],Enochs and Jenda introduced Gorenstein projective modules.The latter are a generalization of the concept of a finitely generated modules of zero G-dimension.In fact,these classes of modules generalize injective and projective modules respectively.Not much later Gorenstein flat modules were introduced and they are easily shown to be an extension of the concept of a flat module.But it was not until 2004 that the definition of Gorenstein modules was given over any associative rings by Holm[60].In 2008,Sean and the coworkers proved that G~2P(R) was the same as the class of Gorenstein projective modules on commutative rings from the analysis of Gorenstein subcategories of abelian categories[94].So,it is natural to study G~2P(R) over noncommutative rings.This is the main goal of Chapter 2,in which weak Gorenstein Gorenstein projective modules are introduced and some homological properties of them are discussed.The theory of covers and envelopes has been an active branch of associative algebra and homological algebra.The existence of a flat cover and a cotorsion envelop for any modules over any associative rings has been proved recently[13].As we all know,module category is a subcategory of the category of complexes of modules.So,it seems to be natural to consider covers and envelopes in the category of complexes of modules.In Chapter 3,we construct a new cotorsion theory for the first time and show that any complex of R-modules has a cover of exact complexes of Gorenstein flat modules over commutive right coherent rings using the theory of covers and envelopes.Strongly Gorenstein flat complexes are also introduced in Chapter 3.It is investigated that,if a complex G is strongly Gorenstein flat,then G is Gorenstein flat;on the other hand,if R is a perfect ring,any Gorenstein projective complex is strongly Gorenstein flat complex.As we know that a complex(G,δ) is injective(projective,flat,respectively) if and only if each R-module Ker(δ~m) is injective(projective,flat,respectively).For strongly Gorenstein flat complexes,we prove that on a left coherent and right perfect ring,if FP-id(_RR)≤n and a complex G is strongly Gorenstein flat,then G~m is strongly Gorenstein flat in _RM for all m∈Z.It is well known,the existence of minimal generator(maximum cogeneraor) is closely correlative with the existence of envelope(cover) with some properties[104].In Chapter 4, we definite the concept of n-Gorenstein injective(we)envelope(n-Gorenstein projective (pre)cover,respectively) and discuss some internal properties of it.Tilting theory plays a central role in the study of the representation theory of artin algebras.It is also important in theory of rings.As a generalization of tilting theory, the theory of star modules were also studied extensively.In 2005,J.Q.Wei gave some characterizations of(not necessarily selfsmall) n-star modules and proved that(not necessarily finitely generated) n-tilting modules were precisely n-star modules n-presenting all the injective.Inspired by this paper,in Chapter 5,we introduce and discuss n-star modules relative to a given class of modules.The main conclusions are a generalization of some results of this reference.