Lemma 59.82.3 (0CAM)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.82: Affine analog of proper base change
Lemma 59.82.3 (cite)

Lemma 59.82.3. Let $Z \subset X$ be a closed subset of a topological space $X$. Assume

$X$ is a spectral space (Topology, Definition 5.23.1), and
for $x \in X$ the intersection $Z \cap \overline{\{ x\} }$ is connected (in particular nonempty).

If $Z = Z_1 \amalg Z_2$ with $Z_ i$ closed in $Z$, then there exists a decomposition $X = X_1 \amalg X_2$ with $X_ i$ closed in $X$ and $Z_ i = Z \cap X_ i$.

Proof. Observe that $Z_ i$ is quasi-compact. Hence the set of points $W_ i$ specializing to $Z_ i$ is closed in the constructible topology by Topology, Lemma 5.24.7. Assumption (2) implies that $X = W_1 \amalg W_2$. Let $x \in \overline{W_1}$. By Topology, Lemma 5.23.6 part (1) there exists a specialization $x_1 \leadsto x$ with $x_1 \in W_1$. Thus $\overline{\{ x\} } \subset \overline{\{ x_1\} }$ and we see that $x \in W_1$. In other words, setting $X_ i = W_ i$ does the job. $\square$

Comments (2)

Comment #3640 by Brian Conrad on October 20, 2018 at 10:48

The reference to Lemma 0902 in the proof is a typo; you meant to refer to Lemma 0903 (and more specifically to part (1) of that Lemma).

Comment #3738 by Johan on October 23, 2018 at 20:03

Thanks and fixed here.

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