The Stacks project

Lemma 37.64.13. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. If $X$, $Y$ are weakly étale over $S$, then $f$ is weakly étale.

Proof. We will use Morphisms, Lemmas 29.25.8 and 29.25.6 without further mention. Write $X \to Y$ as the composition $X \to X \times _ S Y \to Y$. The second morphism is flat as the base change of the flat morphism $X \to S$. The first is the base change of the flat morphism $Y \to Y \times _ S Y$ by the morphism $X \times _ S Y \to Y \times _ S Y$, hence flat. Thus $X \to Y$ is flat. The morphism $X \times _ Y X \to X \times _ S X$ is an immersion. Thus Lemma 37.64.3 implies, that since $X$ is flat over $X \times _ S X$ it follows that $X$ is flat over $X \times _ Y X$. $\square$

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