The following question arises in the context of the proof of Proposition 3.3.18 in Bruns and Herzog, Cohen-Macaulay Rings.

Let $R$ be a CM ring, not necessarily local, and suppose that $R$ admits a canonical module $\omega_R$, i.e. for every maximal ideal $m$ of $R$ we have that $(\omega_R)_m$ is the canonical module of $R_m$. Suppose in addition that

(a) $\omega_R$ has rank equal to $1$

(b) $\omega_R$ is isomorphic to a proper ideal of $R$ (call it also $\omega_R$)

(c) $\dim R >0$.

Let $p$ be a prime ideal of $R$ that contains $\omega_R$. Then Bruns and Herzog write in their proof that

$\operatorname{depth} \frac{R_p}{\omega_R R_p} \ge \operatorname{depth} R_p -1$.

Where does this inequality come from? I suppose B&H use the fact that $\omega_R R_p$ is a maximal CM $R_p$-module, but i still don't see how the inequality follows.


Why don't use Proposition 1.2.9, aka depth lemma? (I leave you the pleasure to find out which one of the three inequalities is suitable here.)


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