The Stacks project

35.25 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 35.25.1. The property $\mathcal{P}(f)=$“$f$ is locally of finite presentation” is fppf local on the source.

Proof. Being locally of finite presentation is Zariski local on the source and the target, see Morphisms, Lemma 29.21.2. It is a property which is preserved under composition, see Morphisms, Lemma 29.21.3. This proves (1), (2) and (3) of Lemma 35.23.4. The final condition (4) is Lemma 35.11.1. Hence we win. $\square$

Lemma 35.25.2. The property $\mathcal{P}(f)=$“$f$ is locally of finite type” is fppf local on the source.

Proof. Being locally of finite type is Zariski local on the source and the target, see Morphisms, Lemma 29.15.2. It is a property which is preserved under composition, see Morphisms, Lemma 29.15.3, and a flat morphism locally of finite presentation is locally of finite type, see Morphisms, Lemma 29.21.8. This proves (1), (2) and (3) of Lemma 35.23.4. The final condition (4) is Lemma 35.11.2. Hence we win. $\square$

Lemma 35.25.3. The property $\mathcal{P}(f)=$“$f$ is open” is fppf local on the source.

Proof. Being an open morphism is clearly Zariski local on the source and the target. It is a property which is preserved under composition, see Morphisms, Lemma 29.23.3, and a flat morphism of finite presentation is open, see Morphisms, Lemma 29.25.10 This proves (1), (2) and (3) of Lemma 35.23.4. The final condition (4) follows from Morphisms, Lemma 29.25.12. Hence we win. $\square$

Lemma 35.25.4. The property $\mathcal{P}(f)=$“$f$ is universally open” is fppf local on the source.

Proof. Let $f : X \to Y$ be a morphism of schemes. Let $\{ X_ i \to X\} _{i \in I}$ be an fppf covering. Denote $f_ i : X_ i \to X$ the compositions. We have to show that $f$ is universally open if and only if each $f_ i$ is universally open. If $f$ is universally open, then also each $f_ i$ is universally open since the maps $X_ i \to X$ are universally open and compositions of universally open morphisms are universally open (Morphisms, Lemmas 29.25.10 and 29.23.3). Conversely, assume each $f_ i$ is universally open. Let $Y' \to Y$ be a morphism of schemes. Denote $X' = Y' \times _ Y X$ and $X'_ i = Y' \times _ Y X_ i$. Note that $\{ X_ i' \to X'\} _{i \in I}$ is an fppf covering also. The morphisms $f'_ i : X_ i' \to Y'$ are open by assumption. Hence by the Lemma 35.25.3 above we conclude that $f' : X' \to Y'$ is open as desired. $\square$


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