Lemma 21.35.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L, M$ be objects of $D(\mathcal{O})$. With the construction as described above there is a canonical isomorphism

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K \otimes _\mathcal {O}^\mathbf {L} L, M)$

in $D(\mathcal{O})$ functorial in $K, L, M$ which recovers (21.35.0.1) on taking $H^0(\mathcal{C}, -)$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet$ representing $M$ and a K-flat complex of $\mathcal{O}$-modules $\mathcal{L}^\bullet$ representing $L$. For any complex of $\mathcal{O}$-modules $\mathcal{K}^\bullet$ we have

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ( \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ), \mathcal{I}^\bullet )$

by Lemma 21.34.1. Note that the left hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M))$ (use Lemma 21.34.8) and that the right hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K \otimes _\mathcal {O}^\mathbf {L} L, M)$. This proves the displayed formula of the lemma. Taking global sections and using Lemma 21.35.1 we obtain (21.35.0.1). $\square$

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