Lemma 52.9.4. If in Lemma 52.9.2 we additionally assume

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

then $H^ i_{J_0}(M) = H^ i_ J(M) = H^ i_\mathfrak m(M)$ for $i \leq s$ and these modules are annihilated by a power of $I$.

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