Lemma 32.8.12 (0EUU)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 32: Limits of Schemes
Section 32.8: Descending properties of morphisms
Lemma 32.8.12 (cite)

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Lemma 32.8.12. Notation and assumptions as in Situation 32.8.1. If

$f$ is an open immersion, and
$f_0$ is locally of finite presentation,

then $f_ i$ is an open immersion for some $i \geq 0$ .

Proof. By Lemma 32.8.10 we can find an $i$ such that $f_ i$ is étale. Then $V_ i = f_ i(X_ i)$ is a quasi-compact open subscheme of $Y_ i$ (Morphisms, Lemma 29.36.13). let $V$ and $V_{i'}$ for $i' \geq i$ be the inverse image of $V_ i$ in $Y$ and $Y_{i'}$ . Then $f : X \to V$ is an isomorphism (namely it is a surjective open immersion). Hence by Lemma 32.8.11 we see that $X_{i'} \to V_{i'}$ is an isomorphism for some $i' \geq i$ as desired. $\square$

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