Lemma 47.9.5 (0ALY)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.9: Local cohomology
Lemma 47.9.5 (cite)

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Lemma 47.9.5. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. For $K, L \in D(A)$ we have

$R\Gamma _ Z(K \otimes _ A^\mathbf {L} L) = K \otimes _ A^\mathbf {L} R\Gamma _ Z(L) = R\Gamma _ Z(K) \otimes _ A^\mathbf {L} L = R\Gamma _ Z(K) \otimes _ A^\mathbf {L} R\Gamma _ Z(L)$

If $K$ or $L$ is in $D_{I^\infty \text{-torsion}}(A)$ then so is $K \otimes _ A^\mathbf {L} L$ .

Proof. By Lemma 47.9.1 we know that $R\Gamma _ Z$ is given by $C \otimes ^\mathbf {L} -$ for some $C \in D(A)$ . Hence, for $K, L \in D(A)$ general we have

$R\Gamma _ Z(K \otimes _ A^\mathbf {L} L) = K \otimes ^\mathbf {L} L \otimes _ A^\mathbf {L} C = K \otimes _ A^\mathbf {L} R\Gamma _ Z(L)$

The other equalities follow formally from this one. This also implies the last statement of the lemma. $\square$

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