The Stacks project

Lemma 48.9.4. With notation as above. For $M \in D(\mathcal{O}_ Z)$ we have

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(Ri_*M, K) = Ri_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Z}(M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, K)) \]

in $D(\mathcal{O}_ Z)$ for all $K$ in $D(\mathcal{O}_ X)$.

Proof. This is immediate from the construction of the functor $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ and the fact that if $\mathcal{K}^\bullet $ is a K-injective complex of $\mathcal{O}_ X$-modules, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \mathcal{K}^\bullet )$ is a K-injective complex of $\mathcal{O}_ Z$-modules, see Derived Categories, Lemma 13.31.9. $\square$

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