Homotopy Equivalences induced by Balanced Pairs

TL;DR

This paper introduces balanced pairs of subcategories in abelian categories and shows they induce equivalences between homotopy categories, with applications to Gorenstein rings and modules.

Contribution

It establishes conditions for balanced pairs to produce homotopy category equivalences and extends known results for Gorenstein rings.

Findings

Homotopy categories of Gorenstein projective and injective modules are equivalent for left-Gorenstein rings.

The equivalence extends Iyengar-Krause's result in the commutative Gorenstein case.

Provides a framework for relating projective and injective modules via balanced pairs.

Abstract

We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which the balanced pair of subcategories gives rise to equivalent 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.