Lemma 48.9.6 (0A79)—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 48.9.6 (cite)

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Lemma 48.9.6. Let $i : Z \to X$ be a closed immersion of schemes. Assume $X$ is a locally Noetherian. Then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ maps $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ into $D^+_{\textit{Coh}}(\mathcal{O}_ Z)$ .

Proof. The question is local on $X$ , hence we may assume that $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(B)$ with $A$ Noetherian and $A \to B$ surjective. In this case, we can apply Lemma 48.9.5 to translate the question into algebra. The corresponding algebra result is a consequence of Dualizing Complexes, Lemma 47.13.4. $\square$

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