Lemma 10.39.6. Let $\{ R_ i, \varphi _{ii'}\} $ be a system of rings over the directed set $I$. Let $R = \mathop{\mathrm{colim}}\nolimits _ i R_ i$.

If $M$ is an $R$-module such that $M$ is flat as an $R_ i$-module for all $i$, then $M$ is flat as an $R$-module.

For $i \in I$ let $M_ i$ be a flat $R_ i$-module and for $i' \geq i$ let $f_{ii'} : M_ i \to M_{i'}$ be a $\varphi _{ii'}$-linear map such that $f_{i' i''} \circ f_{i i'} = f_{i i''}$. Then $M = \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i$ is a flat $R$-module.

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