The Stacks project

Lemma 73.8.1. Let $S$ be a scheme. Let $X \to Y \to Z$ be morphism of algebraic spaces. Let $P$ be one of the following properties of morphisms of algebraic spaces over $S$: flat, locally finite type, locally finite presentation. Assume that $X \to Z$ has $P$ and that $X \to Y$ is a surjection of sheaves on $(\mathit{Sch}/S)_{fppf}$. Then $Y \to Z$ is $P$.

Proof. Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Choose a scheme $V$ and a surjective étale morphism $V \to W \times _ Z Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. By assumption we can find an fppf covering $\{ V_ i \to V\} $ and lifts $V_ i \to X$ of the morphism $V_ i \to Y$. Since $U \to X$ is surjective étale we see that over the members of the fppf covering $\{ V_ i \times _ X U \to V\} $ we have lifts into $U$. Hence $U \to V$ induces a surjection of sheaves on $(\mathit{Sch}/S)_{fppf}$. By our definition of what it means to have property $P$ for a morphism of algebraic spaces (see Morphisms of Spaces, Definition 66.30.1, Definition 66.23.1, and Definition 66.28.1) we see that $U \to W$ has $P$ and we have to show $V \to W$ has $P$. Thus we reduce the question to the case of morphisms of schemes which is treated in Descent, Lemma 35.14.8. $\square$


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