Lemma 10.154.4. Let $I$ be a directed set. Let $i \mapsto (R_ i \to A_ i)$ be a system of arrows of rings over $I$. Set $R = \mathop{\mathrm{colim}}\nolimits R_ i$ and $A = \mathop{\mathrm{colim}}\nolimits A_ i$. If each $A_ i$ is a filtered colimit of étale $R_ i$-algebras, then $A$ is a filtered colimit of étale $R$-algebras.

**Proof.**
This is true because $A = A \otimes _ R R = \mathop{\mathrm{colim}}\nolimits A_ i \otimes _{R_ i} R$ and hence we can apply Lemma 10.154.3 because $R \to A_ i \otimes _{R_ i} R$ is a filtered colimit of étale ring maps by Lemma 10.154.1.
$\square$

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