Lemma 115.4.14. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ be an integer. Assume

1. $A$ has a dualizing complex,

2. if $\mathfrak p \not\in V(I)$ and $V(\mathfrak p) \cap V(I) \not= \{ \mathfrak m\}$, then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim (A/\mathfrak p) > s$.

Then there exists an $n > 0$ and an ideal $J \subset A$ with $V(J) \cap V(I) = \{ \mathfrak m\}$ such that $JI^ n$ annihilates $H^ i_\mathfrak m(M)$ for $i \leq s$.

Proof. According to Local Cohomology, Lemma 51.9.4 we have to show this for the finite $A$-module $E^ i = \text{Ext}^{-i}_ A(M, \omega _ A^\bullet )$ for $i \leq s$. The support $Z$ of $E^0 \oplus \ldots \oplus E^ s$ is closed in $\mathop{\mathrm{Spec}}(A)$ and does not contain any prime as in (2). Hence it is contained in $V(JI^ n)$ for some $J$ as in the statement of the lemma. $\square$

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