Homotopy equivalences induced by balanced pairs

Abstract

We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which a balanced pair of subcategories gives rise to a triangle-equivalence between two homotopy categories of complexes. As an application, we prove that for a left-Gorenstein ring, there exists a triangle-equivalence between the homotopy category of its Gorenstein projective modules and the homotopy category of its Gorenstein injective modules, which restricts to a triangle-equivalence between the homotopy category of projective modules and the homotopy category of injective modules. In the case of commutative Gorenstein rings we prove that up to a natural isomorphism our equivalence extends Iyengar–Krause's equivalence.