## 35.31 Properties of morphisms local in the étale topology on the source

Here are some properties of morphisms that are étale local on the source.

Lemma 35.31.1. The property $\mathcal{P}(f)=$“$f$ is étale” is étale local on the source.

Proof. Combine Lemma 35.26.4 with Morphisms, Lemma 29.36.2 (local for Zariski on source and target), Morphisms, Lemma 29.36.3 (pre-composing), and Lemma 35.14.4 (part (4)). $\square$

Lemma 35.31.2. The property $\mathcal{P}(f)=$“$f$ is locally quasi-finite” is étale local on the source.

Proof. We are going to use Lemma 35.26.4. By Morphisms, Lemma 29.20.11 the property of being locally quasi-finite is local for Zariski on source and target. By Morphisms, Lemmas 29.20.12 and 29.36.6 we see the precomposition of a locally quasi-finite morphism by an étale morphism is locally quasi-finite. Finally, suppose that $X \to Y$ is a morphism of affine schemes and that $X' \to X$ is a surjective étale morphism of affine schemes such that $X' \to Y$ is locally quasi-finite. Then $X' \to Y$ is of finite type, and by Lemma 35.14.2 we see that $X \to Y$ is of finite type also. Moreover, by assumption $X' \to Y$ has finite fibres, and hence $X \to Y$ has finite fibres also. We conclude that $X \to Y$ is quasi-finite by Morphisms, Lemma 29.20.10. This proves the last assumption of Lemma 35.26.4 and finishes the proof. $\square$

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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