Lemma 48.9.4 (0E2I)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.9: Right adjoint of pushforward for closed immersions
Lemma 48.9.4 (cite)

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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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