Lemma 29.36.19 (02K6)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.36: Étale morphisms
Lemma 29.36.19 (cite)

numberstags

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

be a commutative diagram of morphisms of schemes. Assume that

$f$ is surjective, and étale,
$p$ is étale, and
$q$ is locally of finite presentation ¹.

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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