Lemma 35.14.4 (02KM)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.14: Descent of finiteness and smoothness properties of morphisms
Lemma 35.14.4 (cite)

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Lemma 35.14.4. Let

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

be a commutative diagram of morphisms of schemes. Assume that

$f$ is surjective, and syntomic (resp. smooth, resp. étale),
$p$ is syntomic (resp. smooth, resp. étale).

Then $q$ is syntomic (resp. smooth, resp. étale).

Proof. Combine Morphisms, Lemmas 29.30.16, 29.34.19, and 29.36.19 with Lemma 35.14.3 above. $\square$

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