Lemma 66.20.2. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $T \subset |X|$ be an irreducible closed subset. Let $\xi \in T$ be the generic point (Proposition 66.12.4). Then $\text{codim}(T, |X|)$ (Topology, Definition 5.11.1) is the dimension of the local ring of $X$ at $\xi $ (Properties of Spaces, Definition 64.10.2).

**Proof.**
Choose a scheme $U$, a point $u \in U$, and an étale morphism $U \to X$ sending $u$ to $\xi $. Then any sequence of nontrivial specializations $\xi _ e \leadsto \ldots \leadsto \xi _0 = \xi $ can be lifted to a sequence $u_ e \leadsto \ldots \leadsto u_0 = u$ in $U$ by Lemma 66.12.2. Conversely, any sequence of nontrivial specializations $u_ e \leadsto \ldots \leadsto u_0 = u$ in $U$ maps to a sequence of nontrivial specializations $\xi _ e \leadsto \ldots \leadsto \xi _0 = \xi $ by Lemma 66.12.1. Because $|X|$ and $U$ are sober topological spaces we conclude that the codimension of $T$ in $|X|$ and of $\overline{\{ u\} }$ in $U$ are the same. In this way the lemma reduces to the schemes case which is Properties, Lemma 28.10.3.
$\square$

## 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Your email address will not be published.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)