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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