The Stacks project

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.

  1. 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 $U \subset X$,

  2. 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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