Lemma 10.103.10. Suppose $R$ is a Noetherian local ring. Assume there exists a Cohen-Macaulay module $M$ with $\mathop{\mathrm{Spec}}(R) = \text{Supp}(M)$. Then for a prime $\mathfrak p \subset R$ we have

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

Proof. Follows immediately from Lemma 10.103.9. $\square$

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