Lemma 106.10.1. Let Y be a quasi-compact and quasi-separated algebraic space. Let V \subset Y be a quasi-compact open. Let f : \mathcal{X} \to V be surjective, flat, and locally of finite presentation. Then there exists a finite surjective morphism g : Y' \to Y such that V' = g^{-1}(V) \to Y factors Zariski locally through f.
Proof. We first prove this when Y is a scheme. We may choose a scheme U and a surjective smooth morphism U \to \mathcal{X}. Then \{ U \to V\} is an fppf covering of schemes. By More on Morphisms, Lemma 37.48.6 there exists a finite surjective morphism V' \to V such that V' \to V factors Zariski locally through U. By More on Morphisms, Lemma 37.48.4 we can find a finite surjective morphism Y' \to Y whose restriction to V is V' \to V as desired.
If Y is an algebraic space, then we see the lemma is true by first doing a finite base change by a finite surjective morphism Y' \to Y where Y' is a scheme. See Limits of Spaces, Proposition 70.16.1. \square
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