The Stacks project

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

  1. $f$ is an open immersion, and

  2. $f_0$ 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 ├ętale. 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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