Lemma 76.48.8 (0CHL)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.48: Local complete intersection morphisms
Lemma 76.48.8 (cite)

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Lemma 76.48.8. Let $S$ be a scheme. Let

$\xymatrix{ X \ar[rr]_ f \ar[rd] & & Y \ar[ld] \\ & Z }$

be a commutative diagram of morphisms of algebraic spaces over $S$ . Assume $Y \to Z$ is smooth and $X \to Z$ is a local complete intersection morphism. Then $f : X \to Y$ is a local complete intersection morphism.

Proof. Choose a scheme $W$ and a surjective étale morphism $W \to Z$ . Choose a scheme $V$ and a surjective étale morphism $V \to W \times _ Z Y$ . Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$ . Then $U \to W$ is a local complete intersection morphism of schemes and $V \to W$ is a smooth morphism of schemes. By the result for schemes (More on Morphisms, Lemma 37.62.10) we conclude that $U \to V$ is a local complete intersection morphism. By definition this means that $f$ is a local complete intersection morphism. $\square$

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