Map induced by localization on categories

Question

I have been doing some reading in Hartshorne's Algebraic Geometry on derived functors and subsequent results in cohomology. Given $A$ an abelian category of groups, I have seen that the map $\mathrm{Hom}_{K(A)}(K,L) \rightarrow \mathrm{Hom}_{D(A)}(Q(K),Q(L))$ induced by localization $Q:K(A) \rightarrow D(A)$ is, in general, not injective. I suspect that this map is not surjective in general either. Could someone provide an example as to why this might be the case? I think the way to approach the problem would be to show that a quasi-isomorphism from a complex $L$ to another $K$ does not have an inverse in $K(A)$ in general. How might I do this? Any help would be appreciated. Thanks very much.

Answer 1

It is indeed not surjective, for the reason you state. Consider the case of abelian groups: the chain complex $$\cdots \to 0 \to \mathbb{Z} \stackrel{2}{\to} \mathbb{Z} \to 0 \to \cdots$$ is quasi-isomorphic to the chain complex $$\cdots \to 0 \to 0 \to \mathbb{Z} / 2 \mathbb{Z} \to 0 \to \cdots$$ via an obvious chain map, but the only chain map in the reverse direction is the zero map.