The Stacks project

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. $\square$