Lemma 51.7.4 (0BPY)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 51: Local Cohomology
Section 51.7: Finiteness of local cohomology, I
Lemma 51.7.4 (cite)

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Lemma 51.7.4. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Set $Z = V(I)$ . Let $n \geq 0$ be an integer. If $H^ i_ Z(A)$ is finite for $0 \leq i \leq n$ , then the same is true for $H^ i_ Z(M)$ , $0 \leq i \leq n$ for any finite $A$ -module $M$ such that $M_ f$ is a finite locally free $A_ f$ -module for all $f \in I$ .

Proof. The assumption that $H^ i_ Z(A)$ is finite for $0 \leq i \leq n$ implies there exists an $e \geq 0$ such that $I^ e$ annihilates $H^ i_ Z(A)$ for $0 \leq i \leq n$ , see Lemma 51.7.1. Then Lemma 51.7.3 implies that $H^ i_ Z(M)$ , $0 \leq i \leq n$ is annihilated by $I^ m$ for some $m = m(M, i)$ . We may take the same $m$ for all $0 \leq i \leq n$ . Then Lemma 51.7.1 implies that $H^ i_ Z(M)$ is finite for $0 \leq i \leq n$ as desired. $\square$

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