Lemma 10.72.4. Let $R$ be a Noetherian ring, $I \subset R$ an ideal, and $M$ a finite nonzero $R$-module such that $IM \not= M$. Then $\text{depth}_ I(M) < \infty $.
Proof. Since $M/IM$ is nonzero we can choose $\mathfrak p \in \text{Supp}(M/IM)$ by Lemma 10.40.2. Then $(M/IM)_\mathfrak p \not= 0$ which implies $I \subset \mathfrak p$ and moreover implies $M_\mathfrak p \not= IM_\mathfrak p$ as localization is exact. Let $f_1, \ldots , f_ r \in I$ be an $M$-regular sequence. Then $M_\mathfrak p/(f_1, \ldots , f_ r)M_\mathfrak p$ is nonzero as $(f_1, \ldots , f_ r) \subset I$. As localization is flat we see that the images of $f_1, \ldots , f_ r$ form a $M_\mathfrak p$-regular sequence in $I_\mathfrak p$. Since this works for every $M$-regular sequence in $I$ we conclude that $\text{depth}_ I(M) \leq \text{depth}_{I_\mathfrak p}(M_\mathfrak p)$. The latter is $\leq \text{depth}(M_\mathfrak p)$ which is $< \infty $ by Lemma 10.72.3. $\square$
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