The Stacks project

Lemma 32.21.2. Notation and assumptions as in Lemma 32.21.1. Let $f : X \to S$ correspond to $g : Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ via the equivalence. Then $f$ is separated, proper, finite, ├ętale and add more here if and only if $g$ is so.

Proof. The property of being separated, proper, integral, finite, etc is stable under base change. See Schemes, Lemma 26.21.12 and Morphisms, Lemmas 29.41.5 and 29.44.6. Hence if $f$ has the property, then so does $g$. The converse follows from Lemma 32.20.4 but we also give a direct proof here. Namely, if $g$ has to property, then $f$ does in a neighbourhood of $s$ by Lemmas 32.8.6, 32.13.1, 32.8.3, and 32.8.10. Since $f$ clearly has the given property over $S \setminus \{ s\} $ we conclude as one can check the property locally on the base. $\square$

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