The Stacks project

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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