The Stacks project

111.18 Catenary rings

Definition 111.18.1. A Noetherian ring $A$ is said to be catenary if for any triple of prime ideals ${\mathfrak p}_1 \subset {\mathfrak p}_2 \subset {\mathfrak p}_3$ we have

\[ ht({\mathfrak p}_3 / {\mathfrak p}_1) = ht({\mathfrak p}_3/{\mathfrak p}_2) + ht({\mathfrak p}_2/{\mathfrak p}_1). \]

Here $ht(\mathfrak p/\mathfrak q)$ means the height of $\mathfrak p/\mathfrak q$ in the ring $A/\mathfrak q$. In a formula

\[ ht(\mathfrak p/\mathfrak q) = \dim (A_\mathfrak p/\mathfrak qA_\mathfrak p) = \dim ((A/\mathfrak q)_\mathfrak p) = \dim ((A/\mathfrak q)_{\mathfrak p/\mathfrak q}) \]

A topological space $X$ is catenary, if given $T \subset T' \subset X$ with $T$ and $T'$ closed and irreducible, then there exists a maximal chain of irreducible closed subsets

\[ T = T_0 \subset T_1 \subset \ldots \subset T_ n = T' \]

and every such chain has the same (finite) length.

Exercise 111.18.5. Give an example of a finite, sober, catenary topological space $X$ which does not have a dimension function $\delta : X \to \mathbf{Z}$. Here $\delta : X \to \mathbf{Z}$ is a dimension function if for $x, y \in X$ we have

  1. $x \leadsto y$ and $x \not= y$ implies $\delta (x) > \delta (y)$,

  2. $x \leadsto y$ and $\delta (x) \geq \delta (y) + 2$ implies there exists a $z \in X$ with $x \leadsto z \leadsto y$ and $\delta (x) > \delta (z) > \delta (y)$.

Describe your space clearly and succinctly explain why there cannot be a dimension function.


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Your email address will not be published.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 027N. Beware of the difference between the letter 'O' and the digit '0'.