Section 74.19 (06F3): Properties of morphisms local in the étale topology on the source

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74.19 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 74.19.1. The property $\mathcal{P}(f)=$ “ $f$ is étale” is étale local on the source.

Proof. Follows from Lemma 74.14.3 using Morphisms of Spaces, Definition 67.39.1 and Descent, Lemma 35.31.1. $\square$

Lemma 74.19.2. The property $\mathcal{P}(f)=$ “ $f$ is locally quasi-finite” is étale local on the source.

Proof. Follows from Lemma 74.14.3 using Morphisms of Spaces, Definition 67.27.1 and Descent, Lemma 35.31.2. $\square$

Lemma 74.19.3. The property $\mathcal{P}(f)=$ “ $f$ is unramified” is étale local on the source.

Proof. Follows from Lemma 74.14.3 using Morphisms of Spaces, Definition 67.38.1 and Descent, Lemma 35.31.3. $\square$

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74.19 Properties of morphisms local in the étale topology on the source

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