The Stacks project

Proposition 64.13.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Zariski locally quasi-separated (for example if $X$ is quasi-separated), then there exists a dense open subspace $X'$ of $X$ which is a scheme. More precisely, every point $x \in |X|$ of codimension $0$ on $X$ is contained in $X'$.

Proof. The question is local on $X$ by Lemma 64.13.1. Thus by Lemma 64.6.6 we may assume that there exists an affine scheme $U$ and a surjective, quasi-compact, étale morphism $U \to X$. Moreover $U \to X$ is separated (Lemma 64.6.4). Set $R = U \times _ X U$ and denote $s, t : R \to U$ the projections as usual. Then $s, t$ are surjective, quasi-compact, separated, and étale. Hence $s, t$ are also quasi-finite and have finite fibres (Morphisms, Lemmas 29.36.6, 29.20.9, and 29.20.10). By Morphisms, Lemma 29.51.1 for every $\eta \in U$ which is the generic point of an irreducible component of $U$, there exists an open neighbourhood $V \subset U$ of $\eta $ such that $s^{-1}(V) \to V$ is finite. By Descent, Lemma 35.20.23 being finite is fpqc (and in particular étale) local on the target. Hence we may apply More on Groupoids, Lemma 40.6.4 which says that the largest open $W \subset U$ over which $s$ is finite is $R$-invariant. By the above $W$ contains every generic point of an irreducible component of $U$. 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). By construction $s_ W, t_ W : R_ W \to W$ are finite étale. Consider the open subspace $X' = W/R_ W \subset X$ (see Spaces, Lemma 63.10.2). By construction the inclusion map $X' \to X$ induces a bijection on points of codimension $0$. This reduces us to Lemma 64.13.2. $\square$


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