Quillen equivalence for chain homotopy categories induced by balanced pairs
Jiangsheng Hu, Wei Ren, Xiaoyan Yang, Hanyang You
TL;DR
The paper provides a general, model-category-driven criterion for when chain-homotopy categories ${\bf K}(\mathcal{X})$ and ${\bf K}(\mathcal{Y})$ attached to a balanced pair $(\mathcal{X},\mathcal{Y})$ are triangulated equivalent. By realizing ${\bf K}(\mathcal{X})$ and ${\bf K}(\mathcal{Y})$ as homotopy categories of suitable model structures on ${\rm Ch}(\mathcal{A},\mathcal{E})$, the authors derive Quillen equivalences under the natural condition that the corresponding right/left orthogonals coincide, yielding ${\bf K}(\mathcal{X}) \simeq {\bf K}(\mathcal{Y})$. The framework specializes to cotorsion triples and to important module categories, including Gorenstein projective/injective modules and pure projective/injective objects, recovering and extending known results (Chen10, WE24) via a unified model-categorical approach. The methods illuminate how relative derived categories and chain-homotopy categories interrelate, with implications for constructing explicit equivalences in a broad range of abelian and Grothendieck settings. Overall, the work provides a versatile toolkit for establishing triangulated equivalences between chain-homotopy categories arising from balanced pairs and related cotorsion structures.
Abstract
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.