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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