Lemma 35.25.4 (09NQ)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.25: Application of fpqc descent of properties of morphisms
Lemma 35.25.4 (cite)

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Lemma 35.25.4. Let $f : X \to Y$ be a morphism of schemes. Let $X^0$ denote the set of generic points of irreducible components of $X$ . If

$f$ is étale and separated,
for $\xi \in X^0$ we have $\kappa (f(\xi )) = \kappa (\xi )$ , and
if $\xi , \xi ' \in X^0$ , $\xi \not= \xi '$ , then $f(\xi ) \not= f(\xi ')$ ,

then $f$ is an open immersion.

Proof. Immediate from Lemmas 35.25.3 and 35.25.2. $\square$

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