On Balance for Relative Homology

Abstract

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.