Lemma 76.15.2 (0604)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.15: Universal first order thickenings
Lemma 76.15.2 (cite)

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Lemma 76.15.2. Let $S$ be a scheme. Let $Z \to Y \to X$ be morphisms of algebraic spaces over $S$ . If $Z \subset Z'$ is a universal first order thickening of $Z$ over $Y$ and $Y \to X$ is formally étale, then $Z \subset Z'$ is a universal first order thickening of $Z$ over $X$ .

Proof. This is formal. Namely, by Lemma 76.15.1 it suffices to consider solid commutative diagrams (76.15.0.1) with $T'$ an affine scheme. The composition $T \to Z \to Y$ lifts uniquely to $T' \to Y$ as $Y \to X$ is assumed formally étale. Hence the fact that $Z \subset Z'$ is a universal first order thickening over $Y$ produces the desired morphism $a' : T' \to Z'$ . $\square$

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