Let $X$ be a variety. Let $D^b(Coh(X))$ be the derived category of bounded complexes of coherent sheaves on $X$, and $D^b\_{coh}(X)$ be the derived category of bounded complexes of sheaves of $\mathcal{O}\_X$-modules with coherent sheaves as cohomologies. Similarly, we have $D^b\_{qc}(X)$ and $D^b(Qco(X))$ by replacing $coherent$ sheaves to $quasi$-$coherent$ sheaves.


It is proved in "Residues and Duality" by Hartshorne (Chapt. II Corollary 7.19) that $D^b\_{qc}(X)$ and $D^b(Qco(X))$ are derived equivalent. I was wondering if the same thing is still true for $D^b\_{Coh}(X)$ and $D^b(coh(X))$? 


Hartshorne's proof seems could not be generalized to this case because I feel that any quasi-coherent module might not be embedded to a quasi-coherent injective module.



Are $D^b_{coh}(X)$ and $D^b(Coh(X))$ derived equivalent?
For X noetherian this is still true. (Proposition 3.5 in Daniel Huybrechts' book)


