The Stacks project

Lemma 32.8.10. Notation and assumptions as in Situation 32.8.1. If

  1. $f$ is étale,

  2. $f_0$ is locally of finite presentation,

then $f_ i$ is étale for some $i \geq 0$.

Proof. Being étale is local on the source and the target (Morphisms, Lemma 29.36.2) hence we may assume $S_0, X_0, Y_0$ affine (details omitted). The corresponding algebra fact is Algebra, Lemma 10.168.7. $\square$

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