The Stacks project

Proof. Assume $\mathcal{F}_ x$ is a flat $\mathcal{O}_{X, x}$-module for all $x \in X$. In this case, if $\mathcal{G} \to \mathcal{H} \to \mathcal{K}$ is exact, then also $\mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{H} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{K} \otimes _{\mathcal{O}_ X} \mathcal{F}$ is exact because we can check exactness at stalks and because tensor product commutes with taking stalks, see Lemma 17.16.1. Conversely, suppose that $\mathcal{F}$ is flat, and let $x \in X$. Consider the skyscraper sheaves $i_{x, *} M$ where $M$ is a $\mathcal{O}_{X, x}$-module. Note that

again by Lemma 17.16.1. Since $i_{x, *}$ is exact, we see that the fact that $\mathcal{F}$ is flat implies that $M \mapsto M \otimes _{\mathcal{O}_{X, x}} \mathcal{F}_ x$ is exact. Hence $\mathcal{F}_ x$ is a flat $\mathcal{O}_{X, x}$-module. $\square$