I would have to ask for apology for following question by everybody who is familar with algebraic geometry since this might be a quite trivial problem that cause some irritations for me: we consider a morphism $f: Z \to Y$ between schemes. then the induced direct image functor $f_*$ is claimed to be left exact. the proofs I found used always following argument: We consider an exact sequence of sheaves on $Z$
$$0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$$
choose arbitrary open subset $V \subset Y$ and apply firstly $f_*$ and then the $\Gamma(V, -)$ functor. Since $\Gamma(V, -)$ is left exact for sheaves we obtain exact sequence
$$0 \to \mathcal{F}(f^{-1}(V)) \to \mathcal{G}(f^{-1}(V)) \to \mathcal{H}(f^{-1}(V))$$
at this point the proves end. why now we are done? why is this sufficient? I thought that a sequence of sheaves if exact if and only if the induced sequence at every stalk is exact namely we have to verify that
$$0 \to (f_*\mathcal{F})_y \to (f_*\mathcal{G})_y \to (f_*\mathcal{H})_y$$
is exact in every $y \in Y$. at that moment I encounter the problem that I don't know how explicitely calculate the stalk $(f_*\mathcal{F})_y$ of the direct image sheaf. on the other hand the exactness for all sequences of second type seems to be much weaker as the conditions for exactness on stalks as in the last one. or are these two criterions for exactness equivalent?
Best Answer
Left exact on all open sets implies left exact on stalks. This follows from exactness of direct limits for categories of modules, cf. Why do direct limits preserve exactness?.