The Stacks project

Lemma 17.13.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor $\mathcal{H}_ Z : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X|_ Z)$ of Remark 17.13.5 is right adjoint to $i_* : \textit{Mod}(\mathcal{O}_ X|_ Z) \to \textit{Mod}(\mathcal{O}_ X)$.

Proof. We have to show that for any $\mathcal{O}_ X$-module $\mathcal{F}$ and any $\mathcal{O}_ X|_ Z$-module $\mathcal{G}$ we have

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

This is clear because after all any section of $i_*\mathcal{G}$ has support in $Z$. Details omitted. $\square$

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