The Stacks project

Lemma 31.11.8. Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is torsion free if and only if $\mathcal{F}_ x$ is a torsion free $\mathcal{O}_{X, x}$-module for all $x \in X$.

Proof. Omitted. See More on Algebra, Lemma 15.22.6. $\square$

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