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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