The Stacks project

Lemma 15.58.2. Let $R$ be a ring. Let $P^\bullet $ be a complex of $R$-modules. The functors

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

and

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

are exact functors of triangulated categories.

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

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