Let, $ \mathcal F$ be locally free coherent sheaf which is globally generated off a finite set on a surface $S$ .

Then I want to prove that there exists a short exact sequence of the following form :

$0 \to F \to \mathcal F \to E \to 0$

where, $F$ is a globally generated subsheaf of $ \mathcal F$ having isomorphic global sections and $E$ is a $0$ dimensional sheaf supported on points where $\mathcal F$ is not globally generated.

My guess is that the existence of subsheaf has to do with torsion filtration of $\mathcal F$ but I don't see how to obtain it.

Can somebody give any hint(or may be a reference)?

Any help from anyone is welcome


It sounds like $F$ is the subsheaf of $\mathcal{F}$ generated by the global sections of $\mathcal{F}$, i.e. the image of the map $H^0(\mathcal{F}) \otimes \mathcal{O}_S \to \mathcal{F}$.


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