The Stacks project

Lemma 15.104.17. Let $A \to B$ be a ring map. If $A \to B$ is weakly étale, then $A \to B$ induces separable algebraic residue field extensions.

Proof. Let $\mathfrak p$ be a prime of $A$. Then $\kappa (\mathfrak p) \to B \otimes _ A \kappa (\mathfrak p)$ is weakly étale by Lemma 15.104.7. Hence $B \otimes _ A \kappa (\mathfrak p)$ is a filtered colimit of étale $\kappa (\mathfrak p)$-algebras by Lemma 15.104.16. Hence for $\mathfrak q \subset B$ lying over $\mathfrak p$ the extension $\kappa (\mathfrak q)/\kappa (\mathfrak p)$ is a filtered colimit of finite separable extensions by Algebra, Lemma 10.143.4. $\square$

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