The Stacks project

Proposition 66.10.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is reasonable, then there exists a dense open subspace of $X$ which is a scheme.

Proof. By Properties of Spaces, Lemma 64.13.1 the question is local on $X$. Hence we may assume there exists an affine scheme $U$ and a surjective étale morphism $U \to X$ (Properties of Spaces, Lemma 64.6.1). Let $n$ be an integer bounding the degrees of the fibres of $U \to X$ which exists as $X$ is reasonable, see Definition 66.6.1. We will argue by induction on $n$ that whenever

  1. $U \to X$ is a surjective étale morphism whose fibres have degree $\leq n$, and

  2. $U$ is isomorphic to a locally closed subscheme of an affine scheme

then the schematic locus is dense in $X$.

Let $X_ n \subset X$ be the open subspace which is the complement of the closed subspace $Z_{n - 1} \subset X$ constructed in Lemma 66.8.1 using the morphism $U \to X$. Let $U_ n \subset U$ be the inverse image of $X_ n$. Then $U_ n \to X_ n$ is finite locally free of degree $n$. Hence $X_ n$ is a scheme by Properties of Spaces, Proposition 64.14.1 (and the fact that any finite set of points of $U_ n$ is contained in an affine open of $U_ n$, see Properties, Lemma 28.29.5).

Let $X' \subset X$ be the open subspace such that $|X'|$ is the interior of $|Z_{n - 1}|$ in $|X|$ (see Topology, Definition 5.21.1). Let $U' \subset U$ be the inverse image. Then $U' \to X'$ is surjective étale and has degrees of fibres bounded by $n - 1$. By induction we see that the schematic locus of $X'$ is an open dense $X'' \subset X'$. By elementary topology we see that $X'' \cup X_ n \subset X$ is open and dense and we win. $\square$


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