The Stacks project

Lemma 62.11.3. Let $j : U \to X$ be a separated étale morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a torsion coefficient ring. The functor $Rj^! : D(X_{\acute{e}tale}, \Lambda ) \to D(U_{\acute{e}tale}, \Lambda )$ is equal to $j^{-1}$.

Proof. This is true because both $Rj^!$ and $j^{-1}$ are right adjoints to $Rj_! = j_!$. See for example Lemmas 62.11.2 and 62.6.2. $\square$

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