# When is $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ fully faithful?

Let $$\mathcal{A}$$ and $$\mathcal{B}$$ be two abelian categories and let $$\mathcal{F}:\mathcal{A}\to \mathcal{B}$$ be an additive functor. Assume that $$\mathcal{F}$$ is exact and let $$\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$$ denotes its (right and left) derived functor.

Under which assumptions on $$\mathcal{F}$$ is $$\mathcal{D}(\mathcal{F})$$ fully faithful? (Surely, $$\mathcal{F}$$ would be fully faithful, but is it enough?)

Many thanks!

I suspect that it is fully faithful if and only if $$\mathcal F$$ preserves $$Ext^n$$-s between objects. The "only if" part is clearly necessary (see below). I am slightly hesitant about the "if" part, especially for the unbounded categories.
On the other hand, you can write many explicit sufficient conditions for that for bounded or bounded on one side categories. It is slightly harder for unbounded categories but I can think of the following sufficient condition: every object of $$A$$ has an injective envelope in $$B$$, which is also in $$A$$, and $$B$$ has finite injective dimension...
Having $$\mathcal F$$ fully faithful is not enough. Let $$B=mod({\mathbb C}[X])$$ and $$A=mod({\mathbb C}[X]/(X))$$ or $$A=mod({\mathbb C}[X]/(X^2))$$. In both cases, the hom-s in the derived categories are different.