The Stacks project

Lemma 66.20.1. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The following are equivalent

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

  2. for any étale morphism $(Y, y) \to (X, x)$ of pointed algebraic spaces, $y$ is a generic point of an irreducible component of $|Y|$,

  3. for some étale morphism $(Y, y) \to (X, x)$ of pointed algebraic spaces, $y$ is a generic point of an irreducible component of $|Y|$,

  4. the dimension of the local ring of $X$ at $x$ is zero, and

  5. $x$ is a point of codimension $0$ on $X$

Proof. Conditions (4) and (5) are equivalent for any algebraic space by definition, see Properties of Spaces, Definition 64.10.2. Observe that any $Y$ as in (2) and (3) is decent by Lemma 66.6.6. Thus it suffices to prove the equivalence of (1) and (4) as then the equivalence with (2) and (3) follows since the dimension of the local ring of $Y$ at $y$ is equal to the dimension of the local ring of $X$ at $x$. Let $f : U \to X$ be an étale morphism from an affine scheme and let $u \in U$ be a point mapping to $x$.

Assume (1). Let $u' \leadsto u$ be a specialization in $U$. Then $f(u') = f(u) = x$. By Lemma 66.12.1 we see that $u' = u$. Hence $u$ is a generic point of an irreducible component of $U$. Thus $\dim (\mathcal{O}_{U, u}) = 0$ and we see that (4) holds.

Assume (4). The point $x$ is contained in an irreducible component $T \subset |X|$. Since $|X|$ is sober (Proposition 66.12.4) we $T$ has a generic point $x'$. Of course $x' \leadsto x$. Then we can lift this specialization to $u' \leadsto u$ in $U$ (Lemma 66.12.2). This contradicts the assumption that $\dim (\mathcal{O}_{U, u}) = 0$ unless $u' = u$, i.e., $x' = x$. $\square$


Comments (0)


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Your email address will not be published.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 0ABV. Beware of the difference between the letter 'O' and the digit '0'.