Lemma 18.39.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $\{ p_ i\} _{i \in I}$ be a conservative family of points of $\mathcal{C}$. Then $\mathcal{F}$ is flat if and only if $\mathcal{F}_{p_ i}$ is a flat $\mathcal{O}_{p_ i}$-module for all $i \in I$.
Proof. By Lemma 18.39.2 we see one of the implications. For the converse, use that $(\mathcal{F} \otimes _\mathcal {O} \mathcal{G})_ p = \mathcal{F}_ p \otimes _{\mathcal{O}_ p} \mathcal{G}_ p$ by Lemma 18.26.2 (as taking stalks at $p$ is given by $p^{-1}$) and Lemma 18.14.4. $\square$
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