Vanishing of stable homology with respect to a semidualizing module

We investigate stable homology of modules over a commutative noetherian ring R with respect to a semidualzing module C, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces C to be trivial and R to be Gorenstein.

Gorenstein Flat Complexes with Respect to a Semidualizing Module

In this paper, we introduce and study GC-flat complexes over a commutative Noetherian ring, where C is a semidualizing module. We prove that Ge-flat complexes are actually the complexes of Go-flat modules. This complements a result of Yang and Liang. As an application, we get that every complex has a GF-C（C）-cover, where GFC（C） is the class of Ge-flat complexes. We also give a characterization of complexes of modules in HC（FC） that are defined by Sather-Wagstaff, Sharif and White.

Auslander Categories and Free Normalizing Extensions

Let_(R)C_(S) be a semidualizing(R,S)-bimodule.Then_(R)C_(S) induces an equivalent between the Auslander class A_(C)(S)and the Bass class B_C(R).Let A and B be free normalizing extensions of R and S respectively.In this paper,we prove that Hom S(_(B)B_(S),_(R)C_(S))is a semidualizing(A,B)-bimodule under some suitable conditions,and so Hom S(_(B)B_(S),_(R)C_(S))induces an equivalence between the Auslander class AHomS (_(B)B_(S),_(R)C_(S))(B). and the Bass class BHomS (BBS,RCS)(A) Furthermore,under a suitable condition on_(R)C_(S),we develop a generalized Morita theory for Auslander categories.

T_C-Gorenstein Projective,L_C-Gorenstein Injective and H_C-Gorenstein Flat Modules

The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.

V-Gorenstein Injective Modules Preenvelopes and Related Dimension

Let R and S be associative rings and sVR a semidualizing （S-R）-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR（Zv（R）,-） and HomR（-,Zv（R）） exact exact complex . of V-injective modules Ii and Ii,i ∈ N0, such that N We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class ,4v（R） which leads to the fact that V-Gorenstein injective modules admit exact right Iv （R）-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V- Gorenstein injective if and only if N @ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Corenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext Iv （R） （I, N） = 0 for all modules I with finite Iv （R）-injective dimension.

A Study of Tate Homology via the Approximation Theory with Applications to the Depth Formula

In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules.In the second part of the paper we consider a depth equality,called the depth formula,which has been initially introduced by Auslander and developed further by Huneke and Wiegand.As an application of our main result,we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.