Lemma 52.8.4. If in Lemma 52.8.2 we additionally assume
if $\mathfrak p \not\in V(I)$, $\mathfrak p \in T$, then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s$,
then $H^ i_{J_0}(M) = H^ i_ J(M) = H^ i_{J + I}(M)$ for $i \leq s$ and these modules are annihilated by a power of $I$.
Proof. Choose $\mathfrak p \in V(J)$ or $\mathfrak p \in V(J_0)$ but $\mathfrak p \not\in V(J + I) = V(J_0 + I)$. It suffices to show that $H^ i_{\mathfrak pA_\mathfrak p}(M_\mathfrak p) = 0$ for $i \leq s$, see Local Cohomology, Lemma 51.2.6. These groups vanish by condition (6) and Dualizing Complexes, Lemma 47.11.1. The final statement follows from Local Cohomology, Proposition 51.10.1. $\square$
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