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

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

$U$ 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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)