Lemma 76.9.10. The property of being a thickening is fpqc local. Similarly for first order thickenings.

Proof. The statement means the following: Let $S$ be a scheme and let $X \to X'$ be a morphism of algebraic spaces over $S$. Let $\{ g_ i : X'_ i \to X'\}$ be an fpqc covering of algebraic spaces such that the base change $X_ i \to X'_ i$ is a thickening for all $i$. Then $X \to X'$ is a thickening. Since the morphisms $g_ i$ are jointly surjective we conclude that $X \to X'$ is surjective. By Descent on Spaces, Lemma 74.11.17 we conclude that $X \to X'$ is a closed immersion. Thus $X \to X'$ is a thickening. We omit the proof in the case of first order thickenings. $\square$

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