Lemma 10.104.4. Suppose $R$ is a Noetherian local Cohen-Macaulay ring of dimension $d$. For any prime $\mathfrak p \subset R$ we have

$\dim (R) = \dim (R_{\mathfrak p}) + \dim (R/\mathfrak p).$

Proof. Follows immediately from Lemma 10.104.3. (Also, this is a special case of Lemma 10.103.10.) $\square$

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