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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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