Lemma 20.42.9 (0A8U)—The Stacks project

Lemma 20.42.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L, M$ be objects of $D(\mathcal{O}_ X)$ . There is a canonical morphism

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

in $D(\mathcal{O}_ X)$ functorial in $K, L, M$ .

Proof. Choose a K-injective complex $\mathcal{I}^\bullet$ representing $M$ , a K-injective complex $\mathcal{J}^\bullet$ representing $L$ , and a K-flat complex $\mathcal{K}^\bullet$ representing $K$ . The map is defined using the map

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

of Lemma 20.41.5. By our particular choice of complexes the left hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _{\mathcal{O}_ X}^\mathbf {L} K$ and the right hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L), M)$ . We omit the proof that this is functorial in all three objects of $D(\mathcal{O}_ X)$ . $\square$

The Stacks project

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