Lemma 51.10.2. Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be a finite $A$-module, let $\mathfrak p \subset A$ be a prime ideal, and let $s \geq 0$ be an integer. Assume

$A$ has a dualizing complex,

$\mathfrak p \not\in V(I)$, and

for all primes $\mathfrak p' \subset \mathfrak p$ and $\mathfrak q \in V(I)$ with $\mathfrak p' \subset \mathfrak q$ we have

\[ \text{depth}_{A_{\mathfrak p'}}(M_{\mathfrak p'}) + \dim ((A/\mathfrak p')_\mathfrak q) > s \]

Then there exists an $f \in A$, $f \not\in \mathfrak p$ which annihilates $H^ i_{V(I)}(M)$ for $i \leq s$.

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