Lemma 74.8.4 (0AHE)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 74: Descent and Algebraic Spaces
Section 74.8: Descent of finiteness and smoothness properties of morphisms
Lemma 74.8.4 (cite)

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Lemma 74.8.4. Let $S$ be a scheme. Let

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

be a commutative diagram of morphisms of algebraic spaces over $S$ . Assume that

$f$ is surjective, flat, and locally of finite presentation,
$p$ is smooth (resp. étale).

Then $q$ is smooth (resp. étale).

Proof. We deduce this from the analogue for schemes. Namely, the problem is étale local on $B$ and $Y$ (Morphisms of Spaces, Lemmas 67.37.4 and 67.39.2). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 67.30.6), we can choose an affine scheme $U$ and an étale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma 35.14.5. $\square$

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