Let $(X,\mathcal{O}_X)$ be a locally ringed space such that $\mathcal{O}_X$ is locally notherian, and let $\operatorname{Coh}(\mathcal{O}_X)$ be the category of coherent $\mathcal{O}_X$-modules. The inclusion $\operatorname{Coh}(\mathcal{O}_X)\rightarrow \operatorname{Mod}(\mathcal{O}_X)$ induces a map of derived categories:
$$D^b(\operatorname{Coh}(\mathcal{O}_X))\rightarrow D^b_{\operatorname{Coh}}(\operatorname{Mod}(\mathcal{O}_X)),$$ the latter being the derived category of bounded complexes with coherent cohomology.
My question is: When is this map an equivalence?
This holds, for example, whenever $X$ is a noetherian scheme, and more generally if for every epimorphism $\mathcal{G}\rightarrow \mathcal{F}$, where $\mathcal{G}$ is an $\mathcal{O}_X$-module, and $\mathcal{F}$ is coherent, there is a coherent module $\mathcal{E}$ with a map $\mathcal{E}\rightarrow \mathcal{G}$ such that the composition $\mathcal{E}\rightarrow \mathcal{F}$ is still an epimorphism. Are there some conditions on $(X,\mathcal{O}_X)$ that ensure that the map will be an equivalence?
