The Stacks project

Lemma 69.6.10. Notation and assumptions as in Situation 69.6.1. If

  1. $f$ is a isomorphism,

  2. $f_0$ is locally of finite presentation,

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

Proof. Being an isomorphism is equivalent to being ├ętale, universally injective, and surjective, see Morphisms of Spaces, Lemma 66.51.2. Thus the lemma follows from Lemmas 69.6.2, 69.6.4, and 69.6.5. $\square$


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