The Stacks project

Lemma 35.31.3. The property $\mathcal{P}(f)=$“$f$ is unramified” is étale local on the source. The property $\mathcal{P}(f)=$“$f$ is G-unramified” is étale local on the source.

Proof. We are going to use Lemma 35.26.4. By Morphisms, Lemma 29.35.3 the property of being unramified (resp. G-unramified) is local for Zariski on source and target. By Morphisms, Lemmas 29.35.4 and 29.36.5 we see the precomposition of an unramified (resp. G-unramified) morphism by an étale morphism is unramified (resp. G-unramified). Finally, suppose that $X \to Y$ is a morphism of affine schemes and that $f : X' \to X$ is a surjective étale morphism of affine schemes such that $X' \to Y$ is unramified (resp. G-unramified). Then $X' \to Y$ is of finite type (resp. finite presentation), and by Lemma 35.14.2 (resp. Lemma 35.14.1) we see that $X \to Y$ is of finite type (resp. finite presentation) also. By Morphisms, Lemma 29.34.16 we have a short exact sequence

\[ 0 \to f^*\Omega _{X/Y} \to \Omega _{X'/Y} \to \Omega _{X'/X} \to 0. \]

As $X' \to Y$ is unramified we see that the middle term is zero. Hence, as $f$ is faithfully flat we see that $\Omega _{X/Y} = 0$. Hence $X \to Y$ is unramified (resp. G-unramified), see Morphisms, Lemma 29.35.2. This proves the last assumption of Lemma 35.26.4 and finishes the proof. $\square$

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