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Given sheaves $\mathcal{F}_1,\mathcal{F}_2, \mathcal{F}_3$ on some scheme $X$, and an exact sequence

$$ 0\rightarrow \mathcal{F}_1\rightarrow \mathcal{F}_2\rightarrow \mathcal{F}_3\rightarrow 0 ,$$

when do we know that the global section functor $\Gamma(X,*)$ is exact when applied to the above exact sequence? The global section functor is left exact, so I suppose I am asking when does $\Gamma(X,*)$ preserve a surjective morphism of sheaves.

Of course we know this when $X=\operatorname{Spec}(A)$ is an affine scheme and the $\mathcal{F}_i$ are coherent $\mathcal{O}_X$-modules. Are there more notable cases, and more importantly, do we know of an if and only if condition on the scheme and sheaves to ensure that the global section functor is exact?

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2 Answers 2

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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.

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    $\begingroup$ $X$ quasi-compact suffices. $\endgroup$ Mar 1, 2014 at 21:27
  • $\begingroup$ Do we know anything for the case when $X$ is not quasi-compact? $\endgroup$ Mar 1, 2014 at 21:30
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    $\begingroup$ @Martin Thanks. My having learned AG from Hartshorne's book is showing. $\endgroup$
    – SomeEE
    Mar 1, 2014 at 21:37
  • $\begingroup$ Sorry to hijack this thread 4 years later... I thought that if $X$ is quasi-compact, $\mathcal{F}$ quasi-coherent, then $H^i (X, \mathcal{F}) = 0$ for $i$ sufficiently large, but not necessarily for $i>0$. $\endgroup$
    – JJC94
    Aug 11, 2018 at 21:38
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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 $U \subseteq X$.

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