The Stacks project

Lemma 21.17.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{G}^\bullet $ be a complex of $\mathcal{O}$-modules. The functors

\[ K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{G}^\bullet \otimes _\mathcal {O} \mathcal{F}^\bullet ) \]

and

\[ K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{G}^\bullet ) \]

are exact functors of triangulated categories.

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

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