Lemma 10.104.10. Let $(A, \mathfrak m)$ be a Noetherian local ring. The following are equivalent

1. $A$ is catenary, and

2. $\mathfrak p \mapsto \dim (A/\mathfrak p)$ 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 10.104.2). 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. $\square$

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