Lemma 52.9.4. If in Lemma 52.9.2 we additionally assume

1. 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 $I$.

Proof. This is a special case of Lemma 52.8.4. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).