Lemma 48.9.3 (0A77)—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 48.9.3 (cite)

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Lemma 48.9.3. With notation as above. We have

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

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

Proof. This is immediate from the construction of the functor $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ . $\square$

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