Lemma 32.4.10. In Situation 32.4.5. Suppose we are given an $i$ and a locally constructible subset $E \subset S_ i$ such that $f_ i(S) \subset E$. Then $f_{i'i}(S_{i'}) \subset E$ for all sufficiently large $i'$.

**Proof.**
Writing $S_ i$ as a finite union of open affine subschemes reduces the question to the case that $S_ i$ is affine and $E$ is constructible, see Lemma 32.2.2 and Properties, Lemma 28.2.1. In this case the complement $S_ i \setminus E$ is constructible too. Hence there exists an affine scheme $T$ and a morphism $T \to S_ i$ whose image is $S_ i \setminus E$, see Algebra, Lemma 10.29.4. By Lemma 32.4.9 we see that $T \times _{S_ i} S_{i'}$ is empty for all sufficiently large $i'$, and hence $f_{i'i}(S_{i'}) \subset E$ for all sufficiently large $i'$.
$\square$

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## Comments (2)

Comment #5377 by Matthieu Romagny on

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