Let $X$ be a scheme. Let $A$ be a sheaf of associative algebras with 1 on $X$, containing $\mathcal O_X$, which is quasi-coherent as a left and right module over it. Under what conditions is it true that the derived category of quasi-coherent $A$-modules is equal to the derived category of $A$-modules with quasi-coherent cohomology? Do we need to work with bounded below derived category for this? Are there any necessary assumptions on $X$ or $A$ (noetherian?). Either a proof or a reference will be greatly appreciated.