When is the global section functor exact?

Solution 1:

If $X$ is Noetherian, Serre proved that $X$ is affine if and only if $H^i(X,\mathcal{F}) = 0$ for all quasi-coherent $\mathcal{F}$ and $i > 0$.

The latter condition is equivalent to $\Gamma(X,-)$ being an exact functor.

Solution 2:

In addition: One can show that given an exact sequence of sheaves on a topological space $X$ \begin{align*} 0 \rightarrow \mathscr{F}_1 \rightarrow \mathscr{F}_2 \rightarrow \mathscr{F}_3 \rightarrow 0, \end{align*} where $\mathscr{F}_1$ is flasque, the induced sequence \begin{align*} 0 \rightarrow \Gamma(U, \mathscr{F}_1) \rightarrow \Gamma(U, \mathscr{F}_2) \rightarrow \Gamma(U, \mathscr{F}_3) \rightarrow 0 \end{align*} is exact for any open $U \subseteq X$.