Lemma 51.2.3 (0DWS)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 51: Local Cohomology
Section 51.2: Generalities
Lemma 51.2.3 (cite)

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Lemma 51.2.3. Let $I, J \subset A$ be finitely generated ideals of a ring $A$ . If $M$ is an $I$ -power torsion module, then the canonical map

$H^ i_{V(I) \cap V(J)}(M) \to H^ i_{V(J)}(M)$

is an isomorphism for all $i$ .

Proof. Use the spectral sequence of Dualizing Complexes, Lemma 47.9.6 to reduce to the statement $R\Gamma _ I(M) = M$ which is immediate from the construction of local cohomology in Dualizing Complexes, Section 47.9. $\square$

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