# On existence of a short exact sequence of sheaves on surface

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 $$H^0(\mathcal{F}) \otimes \mathcal{O}_S \to \mathcal{F}$$.