The Stacks project

Lemma 32.4.8. In Situation 32.4.5. Suppose for each $i$ we are given a nonempty closed subset $Z_ i \subset S_ i$ with $f_{ii'}(Z_ i) \subset Z_{i'}$. Then there exists a point $s \in S$ with $f_ i(s) \in Z_ i$ for all $i$.

Proof. Let $Z_ i \subset S_ i$ also denote the reduced closed subscheme associated to $Z_ i$, see Schemes, Definition 26.12.5. A closed immersion is affine, and a composition of affine morphisms is affine (see Morphisms, Lemmas 29.11.9 and 29.11.7), and hence $Z_ i \to S_{i'}$ is affine when $i \geq i'$. We conclude that the morphism $f_{ii'} : Z_ i \to Z_{i'}$ is affine by Morphisms, Lemma 29.11.11. Each of the schemes $Z_ i$ is quasi-compact as a closed subscheme of a quasi-compact scheme. Hence we may apply Lemma 32.4.3 to see that $Z = \mathop{\mathrm{lim}}\nolimits _ i Z_ i$ is nonempty. Since there is a canonical morphism $Z \to S$ we win. $\square$


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