Lemma 52.9.2. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume

$A$ has a dualizing complex,

if $\mathfrak p \in V(I)$, then no condition,

if $\mathfrak p \not\in V(I)$ and $V(\mathfrak p) \cap V(I) = \{ \mathfrak m\} $, then $\dim (A/\mathfrak p) \leq d$,

if $\mathfrak p \not\in V(I)$ and $V(\mathfrak p) \cap V(I) \not= \{ \mathfrak m\} $, then

\[ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad \text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim (A/\mathfrak p) > d + s \]

Then there exists an ideal $J_0 \subset A$ with $V(J_0) \cap V(I) = \{ \mathfrak m\} $ such that for any $J \subset J_0$ with $V(J) \cap V(I) = \{ \mathfrak m\} $ the map

induces an isomorphism in cohomology in degrees $\leq s$ and moreover these modules are annihilated by a power of $J_0I$.

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