The Stacks project

Lemma 37.2.6. The property of being a thickening is fpqc local. Similarly for first order thickenings.

Proof. The statement means the following: Let $X \to X'$ be a morphism of schemes and let $\{ g_ i : X'_ i \to X'\} $ be an fpqc covering 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, Lemma 35.20.19 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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