The Stacks project

Lemma 35.25.3. 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 flat 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 universally injective.

Proof. We have to show that $\Delta : X \to X \times _ Y X$ is surjective, see Morphisms, Lemma 29.10.2. As $X \to Y$ is separated, the image of $\Delta $ is closed. Thus if $\Delta $ is not surjective, we can find a generic point $\eta \in X \times _ S X$ of an irreducible component of $X \times _ S X$ which is not in the image of $\Delta $. The projection $\text{pr}_1 : X \times _ Y X \to X$ is flat as a base change of the flat morphism $X \to Y$, see Morphisms, Lemma 29.25.8. Hence generalizations lift along $\text{pr}_1$, see Morphisms, Lemma 29.25.9. We conclude that $\xi = \text{pr}_1(\eta ) \in X^0$. However, assumptions (2) and (3) guarantee that the scheme $(X \times _ Y X)_{f(\xi )}$ has at most one point for every $\xi \in X^0$. In other words, we have $\Delta (\xi ) = \eta $ a contradiction. $\square$

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