Lemma 21.20.8. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let U be an object of \mathcal{C}. Denote j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) the corresponding localization morphism. The restriction functor D(\mathcal{O}) \to D(\mathcal{O}_ U) is a right adjoint to extension by zero j_! : D(\mathcal{O}_ U) \to D(\mathcal{O}).
Proof. We have to show that
\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(j_!E, F) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(E, F|_ U)
Choose a complex \mathcal{E}^\bullet of \mathcal{O}_ U-modules representing E and choose a K-injective complex \mathcal{I}^\bullet representing F. By Lemma 21.20.1 the complex \mathcal{I}^\bullet |_ U is K-injective as well. Hence we see that the formula above becomes
\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(j_!\mathcal{E}^\bullet , \mathcal{I}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{E}^\bullet , \mathcal{I}^\bullet |_ U)
which holds as |_ U and j_! are adjoint functors (Modules on Sites, Lemma 18.19.2) and Derived Categories, Lemma 13.31.2. \square
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