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?


1 Answer 1


For a Noetherian scheme the morphism $$ D^-(\operatorname{Coh}(\mathcal{O}_X))\longrightarrow D^-_{\operatorname{coh}}(\operatorname{Qco}(\mathcal{O}_X)), $$ is an equivalence by SGA6 Exposé II, Proposition 2.2.2, p. 167. The bounded case follows. In the unbounded situation there are counterexamples due to the presence of nilpotent elements in the structure sheaf.

All this is discussed and addressed in a general setting in the paper

Leonid Positselski, Olaf M. Schnürer: Unbounded derived categories of small and big modules: Is the natural functor fully faithful?, Journal of Pure and Applied Algebra, Volume 225, Issue 11, 2021.



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