Lemma 66.4.10. Let S be a scheme. Let X be an algebraic spaces over S. Let U be a scheme and let f : U \to X be an étale morphism. Let X' \subset X be the open subspace corresponding to the open |f|(|U|) \subset |X| via Lemma 66.4.8. Then f factors through a surjective étale morphism f' : U \to X'. Moreover, if R = U \times _ X U, then R = U \times _{X'} U and X' has the presentation X' = U/R.
Proof. The existence of the factorization follows from Lemma 66.4.9. The morphism f' is surjective according to Lemma 66.4.4. To see f' is étale, suppose that T \to X' is a morphism where T is a scheme. Then T \times _ X U = T \times _{X'} U as X' \to X is a monomorphism of sheaves. Thus the projection T \times _{X'} U \to T is étale as we assumed f étale. We have U \times _ X U = U \times _{X'} U as X' \to X is a monomorphism. Then X' = U/R follows from Spaces, Lemma 65.9.1. \square
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