The Stacks project

Lemma 29.36.8. Let $f : X \to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$, then $f$ is étale.

Proof. You can deduce this from Algebra, Lemma 10.143.7. Here is another proof.

By Lemma 29.36.7 a fibre $X_ s$ is étale and hence smooth over $s$. By Lemma 29.34.3 we see that $X \to S$ is smooth. By Lemma 29.35.12 we see that $f$ is unramified. We conclude by Lemma 29.36.5. $\square$

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