The Stacks project

Lemma 70.9.2. Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $x \in |X|$. The following are equivalent

  1. $x$ is a point of codimension $0$ on $X$,

  2. the local ring of $X$ at $x$ has dimension $0$, and

  3. $x$ is a generic point of an irreducible component of $|X|$.

If true, then there exists an open subspace of $X$ containing $x$ which is a scheme.

Proof. The equivalence of (1), (2), and (3) follows from Decent Spaces, Lemma 66.20.1 and the fact that a quasi-separated algebraic space is decent (Decent Spaces, Section 66.6). However in the next paragraph we will give a more elementary proof of the equivalence.

Note that (1) and (2) are equivalent by definition (Properties of Spaces, Definition 64.10.2). To prove the equivalence of (1) and (3) we may assume $X$ is quasi-compact. Choose

\[ \emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X \]

and $f_ i : V_ i \to U_ i$ as in Decent Spaces, Lemma 66.8.6. Say $x \in U_ i$, $x \not\in U_{i + 1}$. Then $x = f_ i(y)$ for a unique $y \in V_ i$. If (1) holds, then $y$ is a generic point of an irreducible component of $V_ i$ (Properties of Spaces, Lemma 64.11.1). Since $f_ i^{-1}(U_{i + 1})$ is a quasi-compact open of $V_ i$ not containing $y$, there is an open neighbourhood $W \subset V_ i$ of $y$ disjoint from $f_ i^{-1}(V_ i)$ (see Properties, Lemma 28.2.2 or more simply Algebra, Lemma 10.25.4). Then $f_ i|_ W : W \to X$ is an isomorphism onto its image and hence $x = f_ i(y)$ is a generic point of $|X|$. Conversely, assume (3) holds. Then $f_ i$ maps $\overline{\{ y\} }$ onto the irreducible component $\overline{\{ x\} }$ of $|U_ i|$. Since $|f_ i|$ is bijective over $\overline{\{ x\} }$, it follows that $\overline{\{ y\} }$ is an irreducible component of $U_ i$. Thus $x$ is a point of codimension $0$.

The final statement of the lemma is Properties of Spaces, Proposition 64.13.3. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 70.9: Schematic locus

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0ADC. Beware of the difference between the letter 'O' and the digit '0'.