Lemma 51.8.8. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Set $Z = V(I)$. Let $M$ be a finite $A$-module. The following are equivalent

1. $H^1_ Z(M)$ is a finite $A$-module, and

2. for all $\mathfrak p \in \text{Ass}(M)$, $\mathfrak p \not\in Z$ and all $\mathfrak q \in V(\mathfrak p + I)$ the completion of $(A/\mathfrak p)_\mathfrak q$ does not have associated primes of dimension $1$.

Proof. Follows immediately from Proposition 51.8.7 via Lemma 51.8.2. $\square$

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