Balance for Relative (Co)Homology in Abelian Categories

Abstract

Let $${\mathcal {A}}$$ be an abelian category, and $${\mathcal {X}}$$ and $${\mathcal {Y}}$$ subcategories. We derive in this paper an additive functor $$T(-, -)$$ using proper $${\mathcal {X}}$$ -resolutions (respectively, proper $${\mathcal {Y}}$$ -coresolutions). Under certain conditions, we establish balance results for such relative homology (respectively, cohomology) over $${\mathcal {A}}$$ . Our main theorem simultaneously recovers theorems of Sather-Wagstaff et al. (J Math Kyoto Univ 48(3):571–596, 2008), Enochs et al. (Relative homological algebra. De Gruyter, Berlin 2000) and Di et al. (Bull Korean Math Soc 52(1):137–147, 2015) as special cases.