Lemma 10.104.3 (00N9)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.104: Cohen-Macaulay rings
Lemma 10.104.3 (cite)

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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. $\square$

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Comment #3038 by Brian Lawrence on December 13, 2017 at 20:48

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

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