Lemma 21.48.1. Let (\mathcal{C}, \mathcal{O}) be a ringed space. The category of complexes of \mathcal{O}-modules with tensor product defined by \mathcal{F}^\bullet \otimes \mathcal{G}^\bullet = \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{G}^\bullet ) is a symmetric monoidal category.
Proof. Omitted. Hints: as unit \mathbf{1} we take the complex having \mathcal{O} in degree 0 and zero in other degrees with obvious isomorphisms \text{Tot}(\mathbf{1} \otimes _\mathcal {O} \mathcal{G}^\bullet ) = \mathcal{G}^\bullet and \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \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
Comments (1)
Comment #10087 by Jhan-Cyuan Syu on