Lemma 70.16.3. Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S such that |X| has finitely many irreducible components.
There exists a surjective finite morphism f : Y \to X of finite presentation where Y is a scheme such that f is finite étale over a quasi-compact dense open U \subset X,
given a surjective étale morphism V \to X we may choose Y \to X such that for every y \in Y there is an open neighbourhood W \subset Y such that W \to X factors through V.
Proof. Part (1) is the special case of (2) with V = X.
Proof of (2). Let \pi : Y \to X be as in Decent Spaces, Lemma 68.9.3 and let U \subset X be a quasi-compact dense open such that \pi ^{-1}(U) \to U is finite étale. Choose a finite affine open covering Y = \bigcup W_ j such that W_ j \to X factors through V. We can write Y = \mathop{\mathrm{lim}}\nolimits Y_ i with \pi _ i : Y_ i \to X finite and of finite presentation such that \pi ^{-1}(U) \to \pi _ i^{-1}(U) is an isomorphism, see Lemma 70.16.2. For large enough i the algebraic space Y_ i is a scheme, see Lemma 70.5.11. For large enough i we can find affine opens W_{i, j} \subset Y_ i whose inverse image in Y recovers W_ j, see Lemma 70.5.7. For even larger i the morphisms W_ j \to V over X come from morphisms W_{i, j} \to U over X, see Proposition 70.3.10. This finishes the proof. \square
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