The Stacks project

Lemma 10.154.5. Let $R$ be a ring. Let $A \to B$ be an $R$-algebra homomorphism. If $A$ and $B$ are filtered colimits of étale $R$-algebras, then $B$ is a filtered colimit of étale $A$-algebras.

Proof. Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ and $B = \mathop{\mathrm{colim}}\nolimits B_ j$ as filtered colimits with $A_ i$ and $B_ j$ étale over $R$. For each $i$ we can find a $j$ such that $A_ i \to B$ factors through $B_ j$, see Lemma 10.127.3. The factorization $A_ i \to B_ j$ is étale by Lemma 10.143.8. Since $A \to A \otimes _{A_ i} B_ j$ is étale (Lemma 10.143.3) it suffices to prove that $B = \mathop{\mathrm{colim}}\nolimits A \otimes _{A_ i} B_ j$ where the colimit is over pairs $(i, j)$ and factorizations $A_ i \to B_ j \to B$ of $A_ i \to B$ (this is a directed system; details omitted). This is clear because colimits commute with tensor products and hence $\mathop{\mathrm{colim}}\nolimits A \otimes _{A_ i} B_ j = A \otimes _ A B = B$. $\square$

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