The Stacks project

Lemma 15.59.14. Let $R$ be a ring. Let $K^\bullet , L^\bullet $ be complexes of $R$-modules. There is a canonical isomorphism

\[ K^\bullet \otimes _ R^\mathbf {L} L^\bullet \longrightarrow L^\bullet \otimes _ R^\mathbf {L} K^\bullet \]

functorial in both complexes which uses a sign of $(-1)^{pq}$ for the map $K^ p \otimes _ R L^ q \to L^ q \otimes _ R K^ p$ (see proof for explanation).

Proof. We may and do replace the complexes by K-flat complexes $K^\bullet $ and $L^\bullet $ and then we use the commutativity constraint discussed in Section 15.58. $\square$

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