I am reading D. Gieseker's On the moduli of vector bundles on an algebraic surface. In Lemma 4.1, the author seems to use the following result
If $0\to G_2\to G_1\to F\to0$ is an exact sequence of coherent sheaves on a smooth surface, and $F$ is torsion free, $G_1$ is locally free, then $G_2$ is locally free.
I wonder whether the above is true and where can I find a reference of such result.
Denote the surface by $S$ and let $x \in S$ be any point. Then we get a short exact sequence on stalks, $$\tag{1} 0 \to G_{2,x} \to G_{1,x} \to F_x \to 0.$$ Since $F$ is torsion-free, so is $F_x$. This means that for any element $a \in \mathcal O_{S,x}$, the map $$F_x \to F_x, f \mapsto a \cdot f$$ is injective. In other words, $a$ is $F_x$-regular. Hence $\operatorname{depth} F_x \geq 1$. Since $G_{1}$ is locally free and $\mathcal O_{S,x}$ is regular, $\operatorname{depth} G_{1,x} = \dim \mathcal O_{S,x}$. Then from (1) we get an estimate $$\operatorname{depth} G_{2,x} \geq \min(\operatorname{depth} G_{1,x}, \operatorname{depth} F_x + 1) = \dim \mathcal O_{S,x}.$$ Here it is crucial that $S$ is a surface, so that $\dim \mathcal O_{S,x} \leq 2 \leq \operatorname{depth} F_x + 1$. So we see that $G_{2,x}$ is free.