Lemma 35.24.1. The property $\mathcal{P}(f) =$“$f$ is an immersion” is fppf local on the base.

**Proof.**
The property of being an immersion is stable under base change, see Schemes, Lemma 26.18.2. The property of being an immersion is Zariski local on the base. Finally, let $\pi : S' \to S$ be a surjective morphism of affine schemes, which is flat and locally of finite presentation. Note that $\pi : S' \to S$ is open by Morphisms, Lemma 29.25.10. Let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is an immersion. In particular we see that $f'(X') = \pi ^{-1}(f(X))$ is locally closed. Hence by Topology, Lemma 5.6.4 we see that $f(X) \subset S$ is locally closed. Let $Z \subset S$ be the closed subset $Z = \overline{f(X)} \setminus f(X)$. By Topology, Lemma 5.6.4 again we see that $f'(X')$ is closed in $S' \setminus Z'$. Hence we may apply Lemma 35.23.19 to the fpqc covering $\{ S' \setminus Z' \to S \setminus Z\} $ and conclude that $f : X \to S \setminus Z$ is a closed immersion. In other words, $f$ is an immersion. Therefore Lemma 35.22.4 applies and we win.
$\square$

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