Derived enhancements of module sheaves on ringed spaces

Assume the Axiom of Choice and fix suitable universes. For any ringed space $(X,{\mathcal{O}}_X)$, the category $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ of all module sheaves is Grothendieck abelian. Its unbounded derived infinity-category $\mathcal{D}(X)$ is stable and presentable and is the infinity-localization of complexes at quasi-isomorphisms. Its homotopy category is the ordinary unbounded derived category.

Import the Grothendieck-category assertion from Stacks 079Q and the derived construction from Higher Algebra at the exact locators above. A ringed space is a ringed site via its category of opens, so the hypothesis applies without quasi-coherence or finiteness. Choice is retained for these resolution/model-category inputs. Their foundational constructions are imported, not reconstructed. The universal property of localization carries degreewise exact additive functors and their natural transformations to the derived localizations; zero objects and mapping cones give exactness. This last application uses the actual functors, rather than only functors on homotopy categories.