Lemma 15.57.17. Let $R$ be a ring. Let $K^\bullet , L^\bullet , M^\bullet$ be complexes of $R$-modules. There is a canonical isomorphism

$(K^\bullet \otimes _ R^\mathbf {L} L^\bullet ) \otimes _ R^\mathbf {L} M^\bullet = K^\bullet \otimes _ R^\mathbf {L} (L^\bullet \otimes _ R^\mathbf {L} M^\bullet )$

functorial in all three complexes.

Proof. Replace the complexes by K-flat complexes and apply Homology, Remark 12.22.8. $\square$

There are also:

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