Models for chain homotopy category of relative acyclic complexes

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Abstract

Let ($\mathcal{X}$, $\mathcal{Y}$) be a balanced pair in an abelian category $\mathcal{A}$. Denote by $\mathbf{K}_{\mathcal{E}-\mathrm{ac}}(\mathcal{X})$ the chain homotopy category of right X-acyclic complexes with all items in $\mathcal{X}$, and dually by $\mathbf{K}_{\mathcal{E}-\mathrm{ac}}(\mathcal{Y})$ the chain homotopy category of left $\mathcal{Y}$-acyclic complexes with all items in $\mathcal{Y}$. We establish realizations of $\mathbf{K}_{\mathcal{E}-\mathrm{ac}}(\mathcal{X})$ and $\mathbf{K}_{\mathcal{E}-\mathrm{ac}}(\mathcal{Y})$ as homotopy categories of model categories under mild conditions. Consequently, we obtain relative versions of recollements of Krause and Neeman-Murfet. We further present applications to Gorenstein projective and Gorenstein injective modules.

Keywords

balanced pair / model category / exact category / chain homotopy category / recollement

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