The Stacks project

Lemma 29.36.19. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume that

  1. $f$ is surjective, and étale,

  2. $p$ is étale, and

  3. $q$ is locally of finite presentation1.

Then $q$ is étale.

Proof. By Lemma 29.34.19 we see that $q$ is smooth. Thus we only need to see that $q$ has relative dimension $0$. This follows from Lemma 29.28.2 and the fact that $f$ and $p$ have relative dimension $0$. $\square$

[1] In fact this is implied by (1) and (2), see Descent, Lemma 35.14.3. Moreover, it suffices to assume that $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma 35.14.5.

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