Lemma 47.10.3 (0957)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.10: Local cohomology for Noetherian rings
Lemma 47.10.3 (cite)

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Lemma 47.10.3. If $A \to B$ is a homomorphism of Noetherian rings and $I \subset A$ is an ideal, then in $D(B)$ we have

$R\Gamma _ I(A) \otimes _ A^\mathbf {L} B = R\Gamma _ Z(A) \otimes _ A^\mathbf {L} B = R\Gamma _ Y(B) = R\Gamma _{IB}(B)$

where $Y = V(IB) \subset \mathop{\mathrm{Spec}}(B)$ .

Proof. Combine Lemmas 47.10.2 and 47.9.3. $\square$

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