Lemma 21.34.2 (0A91)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.34: Hom complexes
Lemma 21.34.2 (cite)

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Lemma 21.34.2. 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

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

of complexes of $\mathcal{O}$ -modules.

Proof. Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 15.71.3. $\square$

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