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The Stacks project

Lemma 21.34.3. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Given complexes \mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet of \mathcal{O}-modules there is a canonical morphism

\text{Tot}\left( \mathcal{K}^\bullet \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \mathcal{L}^\bullet ) \right) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ))

of complexes of \mathcal{O}-modules functorial in all three complexes.

Proof. Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 15.71.4. \square


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