The Stacks project

Lemma 20.26.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{G}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. The functors

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

and

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

are exact functors of triangulated categories.

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

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