The Stacks project

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

  1. 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.

  2. 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.

Proof. Part (1) is a special case of part (2) with $M_ i = M$ for all $i$ and $f_{i i'} = \text{id}_ M$. Proof of (2). Let $\mathfrak a \subset R$ be a finitely generated ideal. By Lemma 10.39.5 it suffices to show that $\mathfrak a \otimes _ R M \to M$ is injective. We can find an $i \in I$ and a finitely generated ideal $\mathfrak a' \subset R_ i$ such that $\mathfrak a = \mathfrak a'R$. Then $\mathfrak a = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathfrak a'R_{i'}$. Since $\otimes $ commutes with colimits the map $\mathfrak a \otimes _ R M \to M$ is the colimit of the maps

\[ \mathfrak a'R_{i'} \otimes _{R_{i'}} M_{i'} \longrightarrow M_{i'} \]

These maps are all injective by assumption. Since colimits over $I$ are exact by Lemma 10.8.8 we win. $\square$

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