The Stacks project

Lemma 64.11.1. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $x \in |X|$. Consider ├ętale morphisms $a : U \to X$ where $U$ is a scheme. The following are equivalent

  1. $x$ is a point of codimension $0$ on $X$,

  2. for some $U \to X$ as above and $u \in U$ with $a(u) = x$, the point $u$ is the generic point of an irreducible component of $U$, and

  3. for any $U \to X$ as above and any $u \in U$ mapping to $x$, the point $u$ is the generic point of an irreducible component of $U$.

If $X$ is representable, this is equivalent to $x$ being a generic point of an irreducible component of $|X|$.

Proof. Observe that a point $u$ of a scheme $U$ is a generic point of an irreducible component of $U$ if and only if $\dim (\mathcal{O}_{U, u}) = 0$ (Properties, Lemma 28.10.4). Hence this follows from the definition of the codimension of a point on $X$ (Definition 64.10.2). $\square$


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