Lemma 10.39.3. Let $R$ be a ring. Let $\{ M_ i, \varphi _{ii'}\} $ be a directed system of flat $R$-modules. Then $\mathop{\mathrm{colim}}\nolimits _ i M_ i$ is a flat $R$-module.

**Proof.**
This follows as $\otimes $ commutes with colimits and because directed colimits are exact, see Lemma 10.8.8.
$\square$

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