The Stacks project

Lemma 10.154.2. Let $A \to B \to C$ be ring maps. If $A \to B$ is a filtered colimit of étale ring maps and $B \to C$ is a filtered colimit of étale ring maps, then $A \to C$ is a filtered colimit of étale ring maps.

Proof. We will use the criterion of Lemma 10.127.4. Let $A \to P \to C$ be a factorization of $A \to C$ with $P$ of finite presentation over $A$. Write $B = \mathop{\mathrm{colim}}\nolimits _{i \in I} B_ i$ where $I$ is a directed set and where $B_ i$ is an étale $A$-algebra. Write $C = \mathop{\mathrm{colim}}\nolimits _{j \in J} C_ j$ where $J$ is a directed set and where $C_ j$ is an étale $B$-algebra. We can factor $P \to C$ as $P \to C_ j \to C$ for some $j$ by Lemma 10.127.3. By Lemma 10.143.3 we can find an $i \in I$ and an étale ring map $B_ i \to C'_ j$ such that $C_ j = B \otimes _{B_ i} C'_ j$. Then $C_ j = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} B_{i'} \otimes _{B_ i} C'_ j$ and again we see that $P \to C_ j$ factors as $P \to B_{i'} \otimes _{B_ i} C'_ j \to C$. As $A \to C' = B_{i'} \otimes _{B_ i} C'_ j$ is étale as compositions and tensor products of étale ring maps are étale. Hence we have factored $P \to C$ as $P \to C' \to C$ with $C'$ étale over $A$ and the criterion of Lemma 10.127.4 applies. $\square$

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