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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_{i'i}(Z_{i'}) \subset Z_ i for all i' \geq 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_{i'i} : 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


Comments (2)

Comment #6811 by 羽山籍真 on

The notation here is not compatible with 086P, where the transition map is from i' to i, but here is from i to i'.


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