MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.