Lemma 18.39.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $p$ be a point of $\mathcal{C}$. If $\mathcal{F}$ is a flat $\mathcal{O}$-module, then $\mathcal{F}_ p$ is a flat $\mathcal{O}_ p$-module.
Proof. In Section 18.37 we have seen that we can think of $p$ as a morphism of ringed topoi
\[ (p, \text{id}_{\mathcal{O}_ p}) : (\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}_ p) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}). \]
such that the pullback functor $p^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ p)$ equals the stalk functor. Thus the lemma follows from Lemma 18.39.1. $\square$
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