Zhenxing Di

On Balance for Relative Homology

Let R be a ring. It is reasonable to use a strict 𝒲𝒳-resolution of a right R-module M of finite 𝒳-projective dimension, where 𝒳 denotes a subcategory of right R-modules closed under extensions and admits an injective cogenerator 𝒲, to define the relative homology functor . A general balance result is established for such relative homology functor that encompasses a theorem of Emmanouil on Gorenstein projective R-modules and extends a theorem of Holm on Gorenstein flat R-modules. We also consider the above relative homology functor with respect to subcategories arising from an arbitrary but fixed semidualizing R-module. Inspired by the idea of Sather-Wagstaff et al. [25], we obtain the corresponding balance results when R is Cohen–Macaulay with a dualizing R-module as applications of our balance result.

Stability of Gorenstein flat categories with respect to a semidualizing module

In this paper, we first introduce

Triangle equivalences involving Gorenstein FP-injective modules

\mathcal {W}_F

Tate homology with respect to cotorsion pairs

-Gorenstein modules to establish the following Foxby equivalence:

Gorenstein AC-projective dimension of unbounded complexes

\xymatrix@C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C(R) \ar@ [r]^{C\otimes_R-} & \mathcal {G}(\mathcal {W}_F) \ar@ [l]^{\textrm{Hom}_R(C,-)}}

Relative derived categories with respect to subcategories

where

BALANCE FOR RELATIVE HOMOLOGY WITH RESPECT TO SEMIDUALIZING MODULES

\mathcal {G}(\mathcal {F})

Relative and Tate homology with respect to semidualizing modules

,

On triangle equivalences of stable categories.

\mathcal {A}_C(R)

Triangulated equivalence between a homotopy category and a triangulated quotient category

and