The Stacks project

Proof. As a first proof one may reduce to the affine case and then use Algebra, Lemma 10.143.8. This proof is somewhat complicated as it uses the “Critère de platitude par fibres” to see that a morphism $X \to Y$ over $S$ between schemes étale over $S$ is automatically flat.

We give a second proof using the graph argument. Namely, consider the factorization $X \to X \times _ S Y \to Y$, where the first arrow is given by $\text{id}_ X$ and $f$ and the second arrow is the projection. We claim both arrows are étale and hence $f$ is étale by Lemma 29.36.3. Namely, the projection is étale as it is the base change of $X \to S$, see Lemma 29.36.4. The first arrow is the base change of the diagonal morphism $Y \to Y \times _ S Y$ because the square

is cartesian. The diagonal $Y \to Y \times _ S Y$ is an open immersion because $Y \to S$ is étale and hence unramified (Lemma 29.36.5) and we may use Lemma 29.35.13. The base change of an open immersion is an open immersion (Schemes, Lemma 26.18.2) and an open immersion is étale (Lemma 29.36.9). This finishes the second proof. $\square$