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The Stacks project

Lemma 73.4.4. 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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