Lemma 48.9.1 (0A75)—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 48.9.1 (cite)

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Lemma 48.9.1. With notation as above. The functor $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ is a right adjoint to the functor $i_* : \textit{Mod}(\mathcal{O}_ Z) \to \textit{Mod}(\mathcal{O}_ X)$ . For $V \subset Z$ open we have

$\Gamma (V, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \mathcal{F})) = \{ s \in \Gamma (U, \mathcal{F}) \mid \mathcal{I}s = 0\}$

where $U \subset X$ is an open whose intersection with $Z$ is $V$ .

Proof. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ Z$ -modules. Then

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{i_*\mathcal{O}_ Z}(i_*\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{O}_ Z, \mathcal{F})) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ Z}(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \mathcal{F}))$

The first equality by Modules, Lemma 17.22.3 and the second by the fully faithfulness of $i_*$ , see Modules, Lemma 17.13.4. The description of sections is left to the reader. $\square$

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