The Stacks project

Lemma 28.2.2. Let $X$ be a scheme and let $E \subset X$ be a locally constructible subset. Let $\xi \in X$ be a generic point of an irreducible component of $X$.

  1. If $\xi \in E$, then an open neighbourhood of $\xi $ is contained in $E$.

  2. If $\xi \not\in E$, then an open neighbourhood of $\xi $ is disjoint from $E$.

Proof. As the complement of a locally constructible subset is locally constructible it suffices to show (2). We may assume $X$ is affine and hence $E$ constructible (Lemma 28.2.1). In this case $X$ is a spectral space (Algebra, Lemma 10.26.2). Then $\xi \not\in E$ implies $\xi \not\in \overline{E}$ by Topology, Lemma 5.23.6 and the fact that there are no points of $X$ different from $\xi $ which specialize to $\xi $. $\square$

Comments (2)

Comment #3440 by on

The statement says is constructible, but the proof seems to work even if is locally constructible. Maybe the statement should say is locally constructible, instead?

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