The Stacks project

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

  1. $f$ is ├ętale and separated,

  2. for $\xi \in X^0$ we have $\kappa (f(\xi )) = \kappa (\xi )$, and

  3. if $\xi , \xi ' \in X^0$, $\xi \not= \xi '$, then $f(\xi ) \not= f(\xi ')$,

then $f$ is an open immersion.

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