Lemma 15.58.4. Let $R$ be a ring. Let $P^\bullet$ be a complex of $R$-modules. The functors

$K(R) \longrightarrow K(R), \quad L^\bullet \longmapsto \text{Tot}(P^\bullet \otimes _ R L^\bullet )$

and

$K(R) \longrightarrow K(R), \quad L^\bullet \longmapsto \text{Tot}(L^\bullet \otimes _ R P^\bullet )$

are exact functors of triangulated categories.

Proof. This follows from Derived Categories, Remark 13.10.9. $\square$

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