The Stacks project

Lemma 32.8.11. Notation and assumptions as in Situation 32.8.1. If

  1. $f$ is an isomorphism, and

  2. $f_0$ is locally of finite presentation,

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

Proof. By Lemmas 32.8.10 and 32.8.5 we can find an $i$ such that $f_ i$ is flat and a closed immersion. Then $f_ i$ identifies $X_ i$ with an open and closed subscheme of $Y_ i$, see Morphisms, Lemma 29.26.2. By assumption the image of $Y \to Y_ i$ maps into $f_ i(X_ i)$. Thus by Lemma 32.4.10 we find that $Y_{i'}$ maps into $f_ i(X_ i)$ for some $i' \geq i$. It follows that $X_{i'} \to Y_{i'}$ is surjective and we win. $\square$


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