This has now been cross-posted: https://mathoverflow.net/questions/404902/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego
It is known that $Coh(\mathcal O_X)\subseteq QCoh(\mathcal O_X)$ is a Serre subcategory for a Noetherian scheme $X$ and $D^-(Coh(\mathcal O_X))\rightarrow D^{-}_{Coh(\mathcal O_X)}(QCoh(\mathcal O_X))$ is an equivalence. A sketch of the proof can be found (for example) in https://stacks.math.columbia.edu/download/perfect.pdf and the proof relies on Lemma 17.4 of https://stacks.math.columbia.edu/download/derived.pdf and the following fact:
For any surjection from a quasi-coherent $O_X$-module $\mathcal G$ to a coherent $\mathcal O_X$-module $\mathcal F$, there is a coherent submodule of $\mathcal G$ which surjects onto $\mathcal F$.
It is later stated that "one can write a quasi-coherent sheaf as the filtered union of its coherent submodules and then one of these will do the job". I think, at least locally, I know what's going on: say $M\rightarrow N$ is a surjection with $N$ a finitely generated module, we can choose the submodule of $M$ (finitely) generated by preimages of the generators of $N$ and this submodule is the "one of these" that finishes the job.
My questions is whether this can be generalied to "nice" situations, for example, let $\mathcal G$ be a locally Noetherian Grothendieck category, consider the Serre subcategory $Noeth(\mathcal G)$, is it true that one always has $D^-(Noeth(\mathcal G))\cong D^-_{Noeth(\mathcal G)}(\mathcal G)$?
As $\mathcal G$ is locally Noetherian, any object is the direct limit of its Noetherian subobjects, is it possible/how to choose a Noetherian subobject that will "do the job"?