The Stacks project

Lemma 31.11.9. Let $X$ be an integral scheme. Let $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ be a short exact sequence of quasi-coherent $\mathcal{O}_ X$-modules. If $\mathcal{F}$ and $\mathcal{F}''$ are torsion free, then $\mathcal{F}'$ is torsion free.

Proof. Omitted. See More on Algebra, Lemma 15.22.5 for the algebraic analogue. $\square$

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