The Stacks project

73.12 Descending properties of morphisms in the fppf topology

In this section we find some properties of morphisms of algebraic spaces for which we could not (yet) show they are local on the base in the fpqc topology which, however, are local on the base in the fppf topology.

Lemma 73.12.1. The property $\mathcal{P}(f) =$“$f$ is an immersion” is fppf local on the base.

Proof. Let $f : X \to Y$ be a morphism of algebraic spaces. Let $\{ Y_ i \to Y\} _{i \in I}$ be an fppf covering of $Y$. Let $f_ i : X_ i \to Y_ i$ be the base change of $f$.

If $f$ is an immersion, then each $f_ i$ is an immersion by Spaces, Lemma 64.12.3. This proves the direct implication in Definition 73.10.1.

Conversely, assume each $f_ i$ is an immersion. By Morphisms of Spaces, Lemma 66.10.7 this implies each $f_ i$ is separated. By Morphisms of Spaces, Lemma 66.27.7 this implies each $f_ i$ is locally quasi-finite. Hence we see that $f$ is locally quasi-finite and separated, by applying Lemmas 73.11.18 and 73.11.24. By Morphisms of Spaces, Lemma 66.51.1 this implies that $f$ is representable!

By Morphisms of Spaces, Lemma 66.12.1 it suffices to show that for every scheme $Z$ and morphism $Z \to Y$ the base change $Z \times _ Y X \to Z$ is an immersion. By Topologies on Spaces, Lemma 72.7.4 we can find an fppf covering $\{ Z_ i \to Z\} $ by schemes which refines the pullback of the covering $\{ Y_ i \to Y\} $ to $Z$. Hence we see that $Z \times _ Y X \to Z$ (which is a morphism of schemes according to the result of the preceding paragraph) becomes an immersion after pulling back to the members of an fppf (by schemes) of $Z$. Hence $Z \times _ Y X \to Z$ is an immersion by the result for schemes, see Descent, Lemma 35.24.1. $\square$

Lemma 73.12.2. The property $\mathcal{P}(f) =$“$f$ is locally separated” is fppf local on the base.

Proof. A base change of a locally separated morphism is locally separated, see Morphisms of Spaces, Lemma 66.4.4. Hence the direct implication in Definition 73.10.1.

Let $\{ Y_ i \to Y\} _{i \in I}$ be an fppf covering of algebraic spaces over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume each base change $X_ i := Y_ i \times _ Y X \to Y_ i$ is locally separated. This means that each of the morphisms

\[ \Delta _ i : X_ i \longrightarrow X_ i \times _{Y_ i} X_ i = Y_ i \times _ Y (X \times _ Y X) \]

is an immersion. The base change of a fppf covering is an fppf covering, see Topologies on Spaces, Lemma 72.7.3 hence $\{ Y_ i \times _ Y (X \times _ Y X) \to X \times _ Y X\} $ is an fppf covering of algebraic spaces. Moreover, each $\Delta _ i$ is the base change of the morphism $\Delta : X \to X \times _ Y X$. Hence it follows from Lemma 73.12.1 that $\Delta $ is a immersion, i.e., $f$ is locally separated. $\square$


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