The Stacks project

35.28 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 35.28.1. The property $\mathcal{P}(f)=$“$f$ is étale” is étale local on the source.

Proof. Combine Lemma 35.23.4 with Morphisms, Lemma 29.36.2 (local for Zariski on source and target), Morphisms, Lemma 29.36.3 (pre-composing), and Lemma 35.11.4 (part (4)). $\square$

Lemma 35.28.2. The property $\mathcal{P}(f)=$“$f$ is locally quasi-finite” is étale local on the source.

Proof. We are going to use Lemma 35.23.4. 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 35.11.2 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 35.23.4 and finishes the proof. $\square$

Lemma 35.28.3. 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 35.23.4. 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 35.11.2 (resp. Lemma 35.11.1) 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 35.23.4 and finishes the proof. $\square$


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