Relative Homology With Respect To Semidualizing Modules

In 1967, the notion of dualizing modules was introduced over a commu-tative Noether ring by Grothendieck, which has extensive applications in commutative algebra and algebraic geometry, especially in the representation theory of algebras and groups. However, to guarantee the existence of dualizing modules, it requires the rings to satisfy more stringent conditions. For instance, over a local Gorenstein （or local Artinian） ring, there exists a dualizing module. As an extension of the notion of dualizing modules, semidualizing modules are abound over an arbitrary ring. Hence, the study of semidu-alizing modules and their related classes of modules has attracted extensive attention of scholars. Based on many previous work of predecessors, in this thesis, we study mainly relative and Tate homology theory with respect to semidualizing modules.In chapter 2, we use a strict WX-resolution of a module M of finite x-projective dimension, where X denotes a subcategory of modules closed under extensions and admits an injective cogenerator W, to define the relative homoiogy functor TorXM{M,-). A general balance result is established for such relative homology functor that encompasses a balance result of Emmanouil on the "Gorenstein" Tor-functor of Gorenstein projective modules and extends a balance result of Holm on Gorenstein flat modules. We also consider the above relative homology functor with respect to subcategories arising from an arbitrary but fixed semidualizing module C, such as TorgPCM（-,-） and TorgFCM（-,-） in which gFc and gPc denote the subcategories of gC-flat and gC-projective modules, respectively. Over a Cohen-Macaulay ring with a dualizing module D, we obtain the corresponding balance results on TorgPCM（-,-） and TorgFCM（-,-） by using resolutions with respect to the semidualizing module Ct=HomR（C. D） as applications of our balance result.In chapter 3, we introduce firstly the notion of WF-Gorenstein modules and explore the relationship with the gC-flat modules. We prove that there exists the following Foxby equivalence: where g（F）, AC and g（Fc） denote the class of Gorenstein flat modules, the Auslander class and the class of WF-Gorenstein modules, respectively. This result refines the classical Foxby equivalence of Auslander class and Bass class from a new perspective. Then, we investigate two-degree WF-Gorenstein modules. A module M is said to be two-degree WF-Gorenstein if there exists an exact sequence G.=…â†'G1â†'G0â†'G1â†'G2â†'… in g（FC） such that M≌Im（G0â†'G₁）, and G. is HomR（g（FC）,-）- and g（FC）*R-exact. We show that two notions of the two-degree WF-Gorenstein and the WF-Gorenstein modules coincide when the ring is commutative GF-closed. The corresponding conclusions of Yang and Bouchiba on Gorenstein flat modules are obtained as corollaries.In chapter 4, we introduce and investigate firstly the notion of Tate.Fc-resolutions of modules over a commutative coherent ring. We show that the class of modules admitting a Tate Fc-resolution is equal to the class of modules of finite g（FC）-projective dimen-sion. Then, we define the Tate homology functor Tate FC-resolutions, and construct the following "A-M type exact sequence" to connect such Tate homology functors and relative homology functors: where the relative homology functor TorFCM （-,-） derived from -（?）- using proper FC-resolutions of the first variable was introduced by Salimi et al. in 2012. This exact sequence of functors provide us a new framework for studying TorFCM（-,-）. Finally, we establish a balance result for such Tate homology over a Cohen-Macaulay ring with a dualizing module by using a good conclusion provided by Enochs et al.In chapter 5, we introduce over an Abel category A the notion of relative derived category with respect to a subcategory X, D*X（S） for * ∈{blank,-,b}, where S is another subcategory of A closed under direct summands. By using the construction of complexes, we obtain the following triangle equivalence: K-（X）≌Dx（res X）. It extends a theorem on triangle equivalences of Gorenstein derived categories established by Gao and Zhang to the bounded below case when X=g（P）. Moreover, we interpret the relative derived functor ExtxA（-,-） in terms of morphisms in such derived category. Finally, as an application, we construct an "A-M type exact sequence" connecting relative cohomology and Tate cohomology functors with respect to a semidualizing module by using the method provided in such derived category.