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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)