Lemma 10.105.10. Let $(A, \mathfrak m)$ be a Noetherian local ring. The following are equivalent

$A$ is catenary, and

$\mathfrak p \mapsto \dim (A/\mathfrak p)$ is a dimension function on $\mathop{\mathrm{Spec}}(A)$.

Lemma 10.105.10. Let $(A, \mathfrak m)$ be a Noetherian local ring. The following are equivalent

$A$ is catenary, and

$\mathfrak p \mapsto \dim (A/\mathfrak p)$ is a dimension function on $\mathop{\mathrm{Spec}}(A)$.

**Proof.**
If $A$ is catenary, then $\mathop{\mathrm{Spec}}(A)$ has a dimension function $\delta $ by Topology, Lemma 5.20.4 (and Lemma 10.105.2). We may assume $\delta (\mathfrak m) = 0$. Then we see that

\[ \delta (\mathfrak p) = \text{codim}(V(\mathfrak m), V(\mathfrak p)) = \dim (A/\mathfrak p) \]

by Topology, Lemma 5.20.2. In this way we see that (1) implies (2). The reverse implication follows from Topology, Lemma 5.20.2 as well. $\square$

Your email address will not be published.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)