Lemma 68.12.2 (03IL)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 68: Decent Algebraic Spaces
Section 68.12: Points on decent spaces
Lemma 68.12.2 (cite)

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Lemma 68.12.2. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$ . Let $x, x' \in |X|$ and assume $x' \leadsto x$ , i.e., $x$ is a specialization of $x'$ . Then for every étale morphism $\varphi : U \to X$ from a scheme $U$ and any $u \in U$ with $\varphi (u) = x$ , exists a point $u'\in U$ , $u' \leadsto u$ with $\varphi (u') = x'$ .

Proof. Combine Lemmas 68.5.1 and 68.7.3. $\square$

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