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, étale 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
Comments (0)