Hello,


When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived categories). It holds when $F$ sends some adopted to $F$ class into an adopted to $G$ class.


My question is whether this is only technical issue, or there is something meaningful behind. For example, is there a situation when this does not hold, and somehow thinking that it does hold give wrong answers. Or, is there a more general context in which it always holds.


Sasha



The composition of derived functors - commutation fails hazardly?
This is far from being a technical issue, there are many examples when it fails. Suppose that A is the category of $\mathbb F\_p$-vector spaces, $B = C$ the category of abelian groups, $F$ the embedding, $G = \mathrm{Hom}(\mathbb Z/p\mathbb Z, -)$. Then it is easy to see that $DF = F$, $D(GF) = GF$, but $DG\circ F \neq G \circ F$, and the equality does not hold.


