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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