The Stacks project

Proof. Let $U \to X$ be an étale surjective morphism, where $U$ is a scheme. Let $R = U \times _ X U$. The open subspaces of $X$ correspond $1 - 1$ with open subschemes of $U$ which are $R$-invariant. Hence there is a set of them. Let $X_ i$, $i \in I$ be the set of open subspaces of $X$ which are schemes, i.e., are representable. Consider the open subspace $X' \subset X$ whose underlying set of points is the open $\bigcup |X_ i|$ of $|X|$. By Lemma 66.4.4 we see that

is a surjective map of sheaves on $(\mathit{Sch}/S)_{fppf}$. But since each $X_ i \to X'$ is representable by open immersions we see that in fact the map is surjective in the Zariski topology. Namely, if $T \to X'$ is a morphism from a scheme into $X'$, then $X_ i \times _{X'} T$ is an open subscheme of $T$. Hence we can apply Schemes, Lemma 26.15.4 to see that $X'$ is a scheme. $\square$

Proof. Let $X' \subset X$ be the maximal open subspace which is a scheme (Lemma 66.13.1). Let $x \in |X|$ be a point of codimension $0$ on $X$. By Lemma 66.11.2 it suffices to show $x \in X'$. Let $U \to X$ be as in the statement of the lemma. 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. By Lemma 66.6.7 the fibre of $|U| \to |X|$ over $x$ is finite, say $\{ \eta _1, \ldots , \eta _ n\} $. By Lemma 66.11.1 each $\eta _ i$ is the generic point of an irreducible component of $U$. By Properties, Lemma 28.29.1 we can find an affine open $W \subset U$ containing $\{ \eta _1, \ldots , \eta _ n\} $ (this is where we use that $U$ is quasi-separated). By Groupoids, Lemma 39.24.1 we may assume that $W$ is $R$-invariant. Since $W \subset U$ is an $R$-invariant affine open, 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 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 65.10.2 and it contains $x$ by construction. $\square$

Proof. The question is local on $X$ by Lemma 66.13.1. Thus by Lemma 66.6.6 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 66.6.4). 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 35.23.23 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 65.10.2). By construction the inclusion map $X' \to X$ induces a bijection on points of codimension $0$. This reduces us to Lemma 66.13.2. $\square$