The Stacks project

Lemma 73.9.4. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\{ f_ i : X_ i \to X\} _{i \in I}$ is a family of morphisms of algebraic spaces with target $X$. Let $U \to X$ be a surjective ├ętale morphism from a scheme towards $X$. Then $\{ f_ i : X_ i \to X\} _{i \in I}$ is an fpqc covering of $X$ if and only if $\{ U \times _ X X_ i \to U\} _{i \in I}$ is an fpqc covering of $U$.

Proof. If $\{ X_ i \to X\} _{i \in I}$ is an fpqc covering, then so is $\{ U \times _ X X_ i \to U\} _{i \in I}$ by Lemma 73.9.3. Assume that $\{ U \times _ X X_ i \to U\} _{i \in I}$ is an fpqc covering. Let $h : Z \to X$ be a morphism from an affine scheme towards $X$. Then we see that $U \times _ X Z \to Z$ is a surjective ├ętale morphism of schemes, in particular open. Hence we can find finitely many affine opens $W_1, \ldots , W_ t$ of $U \times _ X Z$ whose images cover $Z$. For each $j$ we may apply the condition that $\{ U \times _ X X_ i \to U\} _{i \in I}$ is an fpqc covering to the morphism $W_ j \to U$, and obtain a standard fpqc covering $\{ W_{jl} \to W_ j\} $ which refines $\{ W_ j \times _ X X_ i \to W_ j\} _{i \in I}$. Hence $\{ W_{jl} \to Z\} $ is a standard fpqc covering of $Z$ (see Topologies, Lemma 34.9.10) which refines $\{ Z \times _ X X_ i \to X\} $ and we win. $\square$


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