Lemma 37.13.12. Let $X \to Y \to Z$ be morphisms of schemes. Assume $X \to Z$ smooth and $Y \to Z$ étale. Then $X \to Y$ is smooth.

Proof. The morphism $X \to Y$ is locally of finite presentation by Morphisms, Lemma 29.21.11. By Lemma 37.13.7 we have $H^{-1}(\mathop{N\! L}\nolimits _{X/Z}) = 0$ and the module $\Omega _{X/Z}$ is finite locally free. By Lemma 37.13.8 we have $H^{-1}(\mathop{N\! L}\nolimits _{Y/Z}) = H^0(\mathop{N\! L}\nolimits _{Y/Z}) = 0$. By Lemma 37.13.10 we get $H^{-1}(\mathop{N\! L}\nolimits _{X/Y}) = 0$ and $\Omega _{X/Y} \cong \Omega _{X/Z}$ is finite locally free. By Lemma 37.13.7 the morphism $X \to Y$ is smooth. $\square$

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