The Stacks project

Lemma 67.10.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a finite, étale, surjective morphism $U \to X$ where $U$ is a scheme, then there exists a dense open subspace of $X$ which is a scheme.

First proof. The morphism $U \to X$ is finite locally free. Hence there is a decomposition of $X$ into open and closed subspaces $X_ d \subset X$ such that $U \times _ X X_ d \to X_ d$ is finite locally free of degree $d$. Thus we may assume $U \to X$ is finite locally free of degree $d$. In this case, let $U_ i \subset U$, $i \in I$ be the set of affine opens. For each $i$ the morphism $U_ i \to X$ is étale and has universally bounded fibres (namely, bounded by $d$). In other words, $X$ is reasonable and the result follows from Proposition 67.10.1. $\square$

Second proof. The question is local on $X$ (Properties of Spaces, Lemma 65.13.1), hence may assume $X$ is quasi-compact. Then $U$ is quasi-compact. Then there exists a dense open subscheme $W \subset U$ which is separated (Properties, Lemma 28.29.3). Set $Z = U \setminus W$. Let $R = U \times _ X U$ and $s, t : R \to U$ the projections. Then $t^{-1}(Z)$ is nowhere dense in $R$ (Topology, Lemma 5.21.6) and hence $\Delta = s(t^{-1}(Z))$ is an $R$-invariant closed nowhere dense subset of $U$ (Morphisms, Lemma 29.48.7). Let $u \in U \setminus \Delta $ be a generic point of an irreducible component. Since these points are dense in $U \setminus \Delta $ and since $\Delta $ is nowhere dense, it suffices to show that the image $x \in X$ of $u$ is in the schematic locus of $X$. Observe that $t(s^{-1}(\{ u\} )) \subset W$ is a finite set of generic points of irreducible components of $W$ (compare with Properties of Spaces, Lemma 65.11.1). By Properties, Lemma 28.29.1 we can find an affine open $V \subset W$ such that $t(s^{-1}(\{ u\} )) \subset V$. Since $t(s^{-1}(\{ u\} ))$ is the fibre of $|U| \to |X|$ over $x$, we conclude by Lemma 67.10.3. $\square$

Third proof. (This proof is essentially the same as the second proof, but uses fewer references.) Assume $X$ is an algebraic space, $U$ a scheme, and $U \to X$ is a finite étale surjective morphism. Write $R = U \times _ X U$ and denote $s, t : R \to U$ the projections as usual. Note that $s, t$ are surjective, finite and étale. Claim: The union of the $R$-invariant affine opens of $U$ is topologically dense in $U$.

Proof of the claim. Let $W \subset U$ be an affine open. Set $W' = t(s^{-1}(W)) \subset U$. Since $s^{-1}(W)$ is affine (hence quasi-compact) we see that $W' \subset U$ is a quasi-compact open. By Properties, Lemma 28.29.3 there exists a dense open $W'' \subset W'$ which is a separated scheme. Set $\Delta ' = W' \setminus W''$. This is a nowhere dense closed subset of $W''$. Since $t|_{s^{-1}(W)} : s^{-1}(W) \to W'$ is open (because it is étale) we see that the inverse image $(t|_{s^{-1}(W)})^{-1}(\Delta ') \subset s^{-1}(W)$ is a nowhere dense closed subset (see Topology, Lemma 5.21.6). Hence, by Morphisms, Lemma 29.48.7 we see that

\[ \Delta = s\left((t|_{s^{-1}(W)})^{-1}(\Delta ')\right) \]

is a nowhere dense closed subset of $W$. Pick any point $\eta \in W$, $\eta \not\in \Delta $ which is a generic point of an irreducible component of $W$ (and hence of $U$). By our choices above the finite set $t(s^{-1}(\{ \eta \} )) = \{ \eta _1, \ldots , \eta _ n\} $ is contained in the separated scheme $W''$. Note that the fibres of $s$ is are finite discrete spaces, and that generalizations lift along the étale morphism $t$, see Morphisms, Lemmas 29.36.12 and 29.25.9. In this way we see that each $\eta _ i$ is a generic point of an irreducible component of $W''$. Thus, by Properties, Lemma 28.29.1 we can find an affine open $V \subset W''$ such that $\{ \eta _1, \ldots , \eta _ n\} \subset V$. By Groupoids, Lemma 39.24.1 this implies that $\eta $ is contained in an $R$-invariant affine open subscheme of $U$. The claim follows as $W$ was chosen as an arbitrary affine open of $U$ and because the set of generic points of irreducible components of $W \setminus \Delta $ is dense in $W$.

Using the claim we can finish the proof. Namely, if $W \subset U$ is an $R$-invariant affine open, then the restriction $R_ W$ of $R$ to $W$ equals $R_ W = s^{-1}(W) = t^{-1}(W)$ (see Groupoids, Definition 39.19.1 and discussion following it). In particular the maps $R_ W \to W$ are finite étale also. It follows in particular that $R_ W$ is affine. Thus we see that $W/R_ W$ is a scheme, by Groupoids, Proposition 39.23.9. On the other hand, $W/R_ W$ is an open subspace of $X$ by Spaces, Lemma 64.10.2. Hence having a dense collection of points contained in $R$-invariant affine open of $U$ certainly implies that the schematic locus of $X$ (see Properties of Spaces, Lemma 65.13.1) is open dense in $X$. $\square$

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