The Stacks project

[Chapter 0, Proposition 16.5.4, EGA]

Lemma 10.103.7. Let $R$ be a Noetherian local ring. Let $M$ be a finite Cohen-Macaulay $R$-module. If $\mathfrak p \in \text{Ass}(M)$, then $\dim (R/\mathfrak p) = \dim (\text{Supp}(M))$ and $\mathfrak p$ is a minimal prime in the support of $M$. In particular, $M$ has no embedded associated primes.

Proof. By Lemma 10.72.9 we have $\text{depth}(M) \leq \dim (R/\mathfrak p)$. Of course $\dim (R/\mathfrak p) \leq \dim (\text{Supp}(M))$ as $\mathfrak p \in \text{Supp}(M)$ (Lemma 10.63.2). Thus we have equality in both inequalities as $M$ is Cohen-Macaulay. Then $\mathfrak p$ must be minimal in $\text{Supp}(M)$ otherwise we would have $\dim (R/\mathfrak p) < \dim (\text{Supp}(M))$. Finally, minimal primes in the support of $M$ are equal to the minimal elements of $\text{Ass}(M)$ (Proposition 10.63.6) hence $M$ has no embedded associated primes (Definition 10.67.1). $\square$

Comments (2)

Comment #2718 by Tanya Kaushal Srivastava on

The reference of this lemma is EGA IV 16.5.4

Comment #2847 by on

OK, thanks! This is really chapter 0 and not chapter IV. See here for corresponding changes in LaTeX.

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