I have a question about a step in the proof of Prop. 5.2.15 in Liu's "Algebraic Geometry" (page 184):
We have the short exact sequence $0 \to \mathcal{F''} \overset{\alpha}{\rightarrow} \mathcal{F} \overset{\beta}{\rightarrow} \mathcal{F'} \to 0$.
If $s \in \mathcal{F'}(X)$ and $\{U_i\}_i$ is a covering of $X$, why there exist $t_i \in \mathcal{F}(U_i)$ such that $s|_{U_i} = \beta(U_i)(t_i)$?
Or more precisely I know that a ses as above induce ses of stalks in each $x \in X$:
$0 \to \mathcal{F_x''} \overset{\alpha}{\rightarrow} \mathcal{F_x} \overset{\beta}{\rightarrow} \mathcal{F_x'} \to 0$, and also a ses of global section: $0 \to \mathcal{F''}(X) \overset{\alpha}{\rightarrow} \mathcal{F}(X) \overset{\beta}{\rightarrow} \mathcal{F'}(X) \to 0$.
But I'm not sure if it induces also following ses for every open $U_i$:
$0 \to \mathcal{F''}(U_i) \overset{\alpha}{\rightarrow} \mathcal{F}(U_i) \overset{\beta}{\rightarrow} \mathcal{F'}(U_i) \to 0$
wich was obviously required above to find the $t_i$'s.
Liu does not asserts that for any cover $U_i$ there are such sections, but that there exist such cover. Indeed, by definition a sequence of sheaves is exact iff it is exact at every stalk. Now, if $s \in F'(X)$, by exactness at each stalk,for all $x \in X$ there is $t_x$ with $\beta_x(t_x) = s_x$, i.e there is an open $U_x \ni x$, a section $t \in F(U_x)$ with $\beta(t_U)_x = s_x$ : taking the set of all $U_x$ for $x \in X$ give you the cover you were looking for.
Edit : A short exact of sheaves does not induces a short exact of global sections, this is precisely why cohomology is made for ! As an example, let us take the exponential sequence $0 \to \Bbb Z \to \mathcal O \to \mathcal O^* \to 0$ on $X = \Bbb C^*$.Notice that $\text{id}_{\Bbb C^*}$ is a global section of $\mathcal O^*$ but there is no function $g$, with $e^g = id$ (i.e there is no global logarithm on $\Bbb C^*$).