Lemma 63.13.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a largest open subspace $X' \subset X$ which is a scheme.

## 63.13 The schematic locus

Every algebraic space has a largest open subspace which is a scheme; this is more or less clear but we also write out the proof below. Of course this subspace may be empty, for example if $X = \mathbf{A}^1_{\mathbf{Q}}/\mathbf{Z}$ (the universal counter example). On the other hand, if $X$ is for example quasi-separated, then this largest open subscheme is actually dense in $X$!

**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 63.4.4 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. $\square$

In the rest of this section we say that an open subspace $X'$ of an algebraic space $X$ is *dense* if the corresponding open subset $|X'| \subset |X|$ is dense.

Lemma 63.13.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a finite, étale, surjective morphism $U \to X$ where $U$ is a quasi-separated scheme, then there exists a dense open subspace $X'$ of $X$ which is a scheme. More precisely, every point $x \in |X|$ of codimension $0$ in $X$ is contained in $X'$.

**Proof.**
Let $X' \subset X$ be the maximal open subspace which is a scheme (Lemma 63.13.1). Let $x \in |X|$ be a point of codimension $0$ on $X$. By Lemma 63.11.2 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 63.6.7 the fibre of $|U| \to |X|$ over $x$ is finite, say $\{ \eta _1, \ldots , \eta _ n\} $. By Lemma 63.11.1 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 62.10.2 and it contains $x$ by construction.
$\square$

We will improve the following proposition to the case of decent algebraic spaces in Decent Spaces, Theorem 65.10.2.

Proposition 63.13.3. 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 63.13.1. Thus by Lemma 63.6.6 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 63.6.4). 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 29.34.6, 29.19.9, and 29.19.10). By Morphisms, Lemma 29.49.1 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 62.10.2). By construction the inclusion map $X' \to X$ induces a bijection on points of codimension $0$. This reduces us to Lemma 63.13.2.
$\square$

## 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)