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 functor

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

is an exact functor of triangulated categories.

Proof. Omitted. Hint: See More on Algebra, Lemmas 15.57.1 and 15.57.2. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).