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