Quillen equivalence for chain homotopy categories induced by balanced pairs

For a balanced pair $(\mathcal{X},\mathcal{Y})$ in an abelian category, we investigate when the chain homotopy categories ${\bf K}(\mathcal{X})$ and ${\bf K}(\mathcal{Y})$ are triangulated equivalent. To this end, we realize these chain homotopy categories as homotopy categories of certain model categories and give conditions that ensure the existence of a Quillen equivalence between the model categories in question. We further give applications to cotorsion triples, Gorenstein projective and Gorenstein injective modules, as well as pure projective and pure injective objects.

💡 Research Summary

This paper investigates when the chain homotopy categories K(𝓧) and K(𝓨) associated to a balanced pair (𝓧, 𝓨) in an abelian category 𝒜 are triangulated equivalent. A balanced pair means that every object of 𝒜 admits an 𝓧‑resolution which becomes acyclic after applying Hom(–, 𝓨) and a 𝓨‑coresolution which becomes acyclic after applying Hom(𝓧, –). The pair is called admissible when each right 𝓧‑approximation is an epimorphism and each left 𝓨‑approximation is a monomorphism.

The authors first construct an exact structure (𝒜, ℰ) where ℰ consists of those short exact sequences that stay exact after applying Hom(𝓧, –). This makes (𝒜, ℰ) a weakly idempotent‑complete exact category. Passing to complexes, they consider the exact category Ch(𝒜, ℰ) of chain complexes with componentwise ℰ‑exactness.

Two Hovey triples are then defined on Ch(𝒜, ℰ). The first triple \