## 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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