Lemma 36.3.9. Let $X$ be a scheme.

For objects $K, L$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the derived tensor product $K \otimes ^\mathbf {L}_{\mathcal{O}_ X} L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

If $X = \mathop{\mathrm{Spec}}(A)$ is affine then

\[ \widetilde{M^\bullet } \otimes _{\mathcal{O}_ X}^\mathbf {L} \widetilde{K^\bullet } = \widetilde{M^\bullet \otimes _ A^\mathbf {L} K^\bullet } \]for any pair of complexes of $A$-modules $K^\bullet $, $M^\bullet $.

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