Lemma 29.4.3. Let $i : Z \to X$ be a closed immersion of schemes. There is a functor1 $i^! : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Z)$ which is a right adjoint to $i_*$. (Compare Modules, Lemma 17.6.3.)

Proof. Given quasi-coherent $\mathcal{O}_ X$-module $\mathcal{G}$ we consider the subsheaf $\mathcal{H}_ Z(\mathcal{G})$ of $\mathcal{G}$ of local sections annihilated by $\mathcal{I}$. By Lemma 29.4.2 there is a canonical largest quasi-coherent $\mathcal{O}_ X$-submodule $\mathcal{H}_ Z(\mathcal{G})'$. By construction we have

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

for any quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{F}$. Hence we can set $i^!\mathcal{G} = i^*(\mathcal{H}_ Z(\mathcal{G})')$. Details omitted. $\square$

[1] This is likely nonstandard notation.

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