The Stacks project

Lemma 72.4.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\{ X_ i \to X\} _{i \in I}$ be a smooth covering of $X$. Then there exists an étale covering $\{ U_ j \to X\} _{j \in J}$ of $X$ which refines $\{ X_ i \to X\} _{i \in I}$.

Proof. First choose a scheme $U$ and a surjective étale morphism $U \to X$. For each $i$ choose a scheme $W_ i$ and a surjective étale morphism $W_ i \to X_ i$. Then $\{ W_ i \to X\} _{i \in I}$ is a smooth covering which refines $\{ X_ i \to X\} _{i \in I}$. Hence $\{ W_ i \times _ X U \to U\} _{i \in I}$ is a smooth covering of schemes. By More on Morphisms, Lemma 37.38.7 we can choose an étale covering $\{ U_ j \to U\} $ which refines $\{ W_ i \times _ X U \to U\} $. Then $\{ U_ j \to X\} _{j \in J}$ is an étale covering refining $\{ X_ i \to X\} _{i \in I}$. $\square$


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