Lemma 5.20.2. Let $X$ be a topological space. If $X$ is sober and has a dimension function, then $X$ is catenary. Moreover, for any $x \leadsto y$ we have

$\delta (x) - \delta (y) = \text{codim}\left(\overline{\{ y\} }, \ \overline{\{ x\} }\right).$

Proof. Suppose $Y \subset Y' \subset X$ are irreducible closed subsets. Let $\xi \in Y$, $\xi ' \in Y'$ be their generic points. Then we see immediately from the definitions that $\text{codim}(Y, Y') \leq \delta (\xi ) - \delta (\xi ') < \infty$. In fact the first inequality is an equality. Namely, suppose

$Y = Y_0 \subset Y_1 \subset \ldots \subset Y_ e = Y'$

is any maximal chain of irreducible closed subsets. Let $\xi _ i \in Y_ i$ denote the generic point. Then we see that $\xi _ i \leadsto \xi _{i + 1}$ is an immediate specialization. Hence we see that $e = \delta (\xi ) - \delta (\xi ')$ as desired. This also proves the last statement of the lemma. $\square$

## 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Your email address will not be published.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.

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 02IA. Beware of the difference between the letter 'O' and the digit '0'.