The Stacks project

Lemma 67.13.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ with residual space $Z_ x \subset X$. Assume $X$ is locally Noetherian. Then $x$ is a closed point of $|X|$ if and only if the morphism $Z_ x \to X$ is a closed immersion.

Proof. If $Z_ x \to X$ is a closed immersion, then $x$ is a closed point of $|X|$, see Morphisms of Spaces, Lemma 66.12.3. Conversely, assume $x$ is a closed point of $|X|$. Let $Z \subset X$ be the reduced closed subspace with $|Z| = \{ x\} $ (Properties of Spaces, Lemma 65.12.3). Then $Z$ is locally Noetherian by Morphisms of Spaces, Lemmas 66.23.7 and 66.23.5. Since also $Z$ is reduced and $|Z| = \{ x\} $ it $Z = Z_ x$ is the residual space by definition. $\square$


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