Lemma 51.8.8 (0BL9)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 51: Local Cohomology
Section 51.8: Finiteness of pushforwards, I
Lemma 51.8.8 (cite)

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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

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