Lemma 20.33.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $j : U \to X$ be an open subspace. Let $T \subset X$ be a closed subset contained in $U$.

If $E$ is an object of $D(\mathcal{O}_ X)$ whose cohomology sheaves are supported on $T$, then $E \to Rj_*(E|_ U)$ is an isomorphism.

If $F$ is an object of $D(\mathcal{O}_ U)$ whose cohomology sheaves are supported on $T$, then $j_!F \to Rj_*F$ is an isomorphism.

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