Lemma 52.8.3. In Lemma 52.8.2 if instead of the empty condition (2) we assume
if \mathfrak p \in V(I), \mathfrak p \not\in V(J) \cap V(I), then \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > s for all \mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I),
then the conditions also imply that H^ i_{J_0}(M) is a finite A-module for i \leq s.
Proof. Recall that H^ i_{J_0}(M) = H^ i_ T(M), see proof of Lemma 52.8.2. Thus it suffices to check that for \mathfrak p \not\in T and \mathfrak q \in T with \mathfrak p \subset \mathfrak q we have \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > s, see Local Cohomology, Proposition 51.11.1. Condition (2') tells us this is true for \mathfrak p \in V(I). Since we know H^ i_ T(M) is annihilated by a power of IJ_0 we know the condition holds if \mathfrak p \not\in V(IJ_0) by Local Cohomology, Proposition 51.10.1. This covers all cases and the proof is complete. \square
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