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. $\square$
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