Lemma 20.42.2. Let (X, \mathcal{O}_ X) be a ringed space. Let K, L, M be objects of D(\mathcal{O}_ X). 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}_ X}^\mathbf {L} L, M)
in D(\mathcal{O}_ X) functorial in K, L, M which recovers (20.42.0.1) by taking H^0(X, -).
Proof.
Choose a K-injective complex \mathcal{I}^\bullet representing M and a K-flat complex of \mathcal{O}_ X-modules \mathcal{L}^\bullet representing L. Let \mathcal{K}^\bullet be any complex of \mathcal{O}_ X-modules representing K. Then 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}_ X} \mathcal{L}^\bullet ), \mathcal{I}^\bullet )
by Lemma 20.41.1. 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 20.41.8) and that the right hand side represents R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} L, M). This proves the displayed formula of the lemma. Taking global sections and using Lemma 20.42.1 we obtain (20.42.0.1).
\square
Comments (2)
Comment #6719 by Bach on
Comment #6915 by Johan on
There are also: