Applications of Balanced Pairs

Abstract

Let $(\mathcal{X}$, $\mathcal{Y})$ be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to $(\mathcal{X}$, $\mathcal{Y})$, and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the $\mathcal{X}$-resolution dimension of $\mathcal{Y}$ (resp. $\mathcal{Y}$-coresolution dimension of $\mathcal{X}$) is finite, then the bounded homotopy category of $\mathcal{Y}$ (resp. $\mathcal{X}$) is contained in that of $\mathcal{X}$ (resp. $\mathcal{Y}$). As a consequence, we get that the right $\mathcal{X}$-singularity category coincides with the left $\mathcal{Y}$-singularity category if the $\mathcal{X}$-resolution dimension of $\mathcal{Y}$ and the $\mathcal{Y}$-coresolution dimension of $\mathcal{X}$ are finite.

17 pages, accepted for publication in Science China Mathematics