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Let $F:{\cal A}\to {\cal B}$ be an additive, exact and faithful functor between abelian categories. Then on the level of complexes, $F$ maps quasi-isomorphisms to quasi-isomorphisms and thus induces a functor on the derived categories $DF:D({\cal A})\to D({\cal B})$.

Is it true that $DF$ is faithful? Likewise for the usual categories of bounded complexes. If not, are there extra conditions that guarantee faithfulness?

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up vote 9 down vote accepted

This will usually not be the case. For example, consider a "typical forgetful functor" for example

$$A-Mod \rightarrow k-vect$$

from the category of representations of a k-algebra $A$ to vectorspaces. It is exact faithful, but its derived functor will only be faithful if the algebra is semi-simple, since there are no Ext between vector spaces. For a concrete example take $A=k[x]$.

More philosophically, I think of an exact faithful functor as a functor which forgets some additional structure. The question whether $DF$ is faithful is the question whether there are no new extensions coming from the additional structure. This will rarely be the case.

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