Proposition 70.16.1. Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S.
There exists a surjective finite morphism Y \to X of finite presentation where Y is a scheme,
given a surjective étale morphism U \to X we may choose Y \to X such that for every y \in Y there is an open neighbourhood V \subset Y such that V \to X factors through U.
Proof. Part (1) is the special case of (2) with U = X. Let Y \to X be as in Decent Spaces, Lemma 68.9.2. Choose a finite affine open covering Y = \bigcup V_ j such that V_ j \to X factors through U. We can write Y = \mathop{\mathrm{lim}}\nolimits Y_ i with Y_ i \to X finite and of finite presentation, see Lemma 70.11.2. 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 V_{i, j} \subset Y_ i whose inverse image in Y recovers V_ j, see Lemma 70.5.7. For even larger i the morphisms V_ j \to U over X come from morphisms V_{i, j} \to U over X, see Proposition 70.3.10. This finishes the proof. \square
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