Preresolving categories and derived equivalences

Abstract

It is well known that a resolving subcategory $\mathcal{A}$ of an abelian subcategory $\mathcal{E}$ induces several derived equivalences: a triangle equivalence $\mathbf{D}^-(\mathcal{A})\to \mathbf{D}^-(\mathcal{E})$ exists in general and furthermore restricts to a triangle equivalence $\mathbf{D}^{\mathsf{b}}(\mathcal{A})\to \mathbf{D}^{\mathsf{b}}(\mathcal{E})$ if $\operatorname{res.dim}_{\mathcal{A}}(E)<\infty$ for any object $E\in \mathcal{E}$. If the category $\mathcal{E}$ is uniformly bounded, i.e. $\operatorname{res.dim}_{\mathcal{A}}(\mathcal{E})<\infty$, one obtains a triangle equivalence $\mathbf{D}(\mathcal{A})\to \mathbf{D}(\mathcal{E})$. In this paper, we show that all of the above statements hold for preresolving subcategories of (one-sided) exact categories. By passing to a one-sided language, one can remove the assumption that $\mathcal{A}\subseteq \mathcal{E}$ is extension-closed completely from the classical setting, yielding easier criteria and more examples. To illustrate this point, we consider the Isbell category $\mathcal{I}$ and show that $\mathcal{I}\subseteq \mathsf{Ab}$ is preresolving but $\mathcal{I}$ cannot be realized as an extension-closed subcategory of an exact category. We also consider a criterion given by Keller to produce derived equivalences of fully exact subcategories. We show that this criterion fits into the framework of preresolving subcategories by considering the relative weak idempotent completion of said subcategory.