Lemma 15.57.2. Let $R$ be a ring. Let $P^\bullet $ be a complex of $R$-modules. The functor

is an exact functor of triangulated categories.

Lemma 15.57.2. Let $R$ be a ring. Let $P^\bullet $ be a complex of $R$-modules. The functor

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

is an exact functor of triangulated categories.

**Proof.**
By our definition of the triangulated structure on $K(\text{Mod}_ R)$ we have to check that our functor maps a termwise split short exact sequence of complexes to a termwise split short exact sequence of complexes. As the terms of $\text{Tot}(L^\bullet \otimes _ R P^\bullet )$ are direct sums of the tensor products $L^ a \otimes _ R P^ b$ this is clear.
$\square$

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