Lemma 10.104.3. Let $R$ be Noetherian local. Suppose $R$ is Cohen-Macaulay of dimension $d$. Any maximal chain of ideals $\mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_ n$ has length $n = d$.

Proof. Special case of Lemma 10.103.9. $\square$

Comment #3038 by Brian Lawrence on

Suggested slogan: In a Cohen-Macaulay ring, any maximal chain of prime ideals has length equal to the dimension.

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