Lemma 24.19.2. In the situation above, let $\mathcal{M}$ be a right differential graded $\mathcal{A}_ U$-module and let $\mathcal{N}$ be a left differential graded $\mathcal{A}$-module. Then

$j_!\mathcal{M} \otimes _\mathcal {A} \mathcal{N} = j_!(\mathcal{M} \otimes _{\mathcal{A}_ U} \mathcal{N}|_ U)$

as complexes of $\mathcal{O}$-modules functorially in $\mathcal{M}$ and $\mathcal{N}$.

Proof. As graded modules, this follows from Lemma 24.10.2. We omit the verification that this isomorphism is compatible with differentials. $\square$

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