Lemma 20.50.1. Let (X, \mathcal{O}_ X) be a ringed space. The category of complexes of \mathcal{O}_ X-modules with tensor product defined by \mathcal{F}^\bullet \otimes \mathcal{G}^\bullet = \text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{G}^\bullet ) is a symmetric monoidal category (for sign rules, see More on Algebra, Section 15.72).
Proof. Omitted. Hints: as unit \mathbf{1} we take the complex having \mathcal{O}_ X in degree 0 and zero in other degrees with obvious isomorphisms \text{Tot}(\mathbf{1} \otimes _{\mathcal{O}_ X} \mathcal{G}^\bullet ) = \mathcal{G}^\bullet and \text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathbf{1}) = \mathcal{F}^\bullet . to prove the lemma you have to check the commutativity of various diagrams, see Categories, Definitions 4.43.1 and 4.43.9. The verifications are straightforward in each case. \square
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