Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also? 


I would be interested in any necessary or sufficient conditions, and also in references! 



When an exact embedding of abelian categories induces a full embedding of their derived categories?
It suffices to require that through any epimorphism in $A'$ from an object of $A'$ onto an object of $A$ some epimorphism in $A$ (onto the same object) would factorize; or the dual condition for monomorphisms. The natural generality is that of exact categories (or perhaps even wider, but at least so).


The standard reference is Keller "Derived categories and their uses", Section 12. Some additional details are recorded in my preprint "Contraherent cosheaves", <http://arxiv.org/abs/1209.2995v3> , Proposition A.2.1 (no originality claim presumed).


