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Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?

Let $i: Z \rightarrow X$ be a closed immersion of schemes. Then, for any $\mathcal{O}_{Z}$-module $\mathcal{G}$, the counit of adjunction $i^{*}i_{*}\mathcal{G} \rightarrow \mathcal{G}$ is an ...
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