The Stacks project

Lemma 52.8.3. In Lemma 52.8.2 if instead of the empty condition (2) we assume

  1. 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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