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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