The Stacks project

Lemma 67.7.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \to X$ be an ├ętale morphism from a scheme to $X$. Assume $u, u' \in |U|$ map to the same point $x$ of $|X|$, and $u' \leadsto u$. If $X$ is locally Noetherian, then $u = u'$.

Proof. The discussion in Schemes, Section 26.13 shows that $\mathcal{O}_{U, u'}$ is a localization of the Noetherian local ring $\mathcal{O}_{U, u}$. By Properties of Spaces, Lemma 65.10.1 we have $\dim (\mathcal{O}_{U, u}) = \dim (\mathcal{O}_{U, u'})$. By dimension theory for Noetherian local rings we conclude $u = u'$. $\square$

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