The Stacks project

72.15 Properties of morphisms local in the fppf topology on the source

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

Lemma 72.15.1. The property $\mathcal{P}(f)=$“$f$ is locally of finite presentation” is fppf local on the source.

Proof. Follows from Lemma 72.13.3 using Morphisms of Spaces, Definition 65.28.1 and Descent, Lemma 35.25.1. $\square$

Lemma 72.15.2. The property $\mathcal{P}(f)=$“$f$ is locally of finite type” is fppf local on the source.

Proof. Follows from Lemma 72.13.3 using Morphisms of Spaces, Definition 65.23.1 and Descent, Lemma 35.25.2. $\square$

Lemma 72.15.3. The property $\mathcal{P}(f)=$“$f$ is open” is fppf local on the source.

Proof. Follows from Lemma 72.13.3 using Morphisms of Spaces, Definition 65.6.2 and Descent, Lemma 35.25.3. $\square$

Lemma 72.15.4. The property $\mathcal{P}(f)=$“$f$ is universally open” is fppf local on the source.

Proof. Follows from Lemma 72.13.3 using Morphisms of Spaces, Definition 65.6.2 and Descent, Lemma 35.25.4. $\square$


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