The Stacks project

Lemma 66.7.4. Let $\mathcal{P}$ be a property of germs of schemes which is ├ętale local, see Descent, Definition 35.21.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point of $X$. Consider ├ętale morphisms $a : U \to X$ where $U$ is a scheme. The following are equivalent

  1. for any $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$, and

  2. for some $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$.

If $X$ is representable, then this is equivalent to $\mathcal{P}(X, x)$.

Proof. Omitted. $\square$

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