Lemma 20.32.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $U \subset X$ be an open subset. Denote $j : (U, \mathcal{O}_ U) \to (X, \mathcal{O}_ X)$ the corresponding open immersion. The restriction functor $D(\mathcal{O}_ X) \to D(\mathcal{O}_ U)$ is a right adjoint to extension by zero $j_! : D(\mathcal{O}_ U) \to D(\mathcal{O}_ X)$.

Proof. This follows formally from the fact that $j_!$ and $j^*$ are adjoint and exact (and hence $Lj_! = j_!$ and $Rj^* = j^*$ exist), see Derived Categories, Lemma 13.30.3. $\square$

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