Lemma 20.34.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $Z \subset X$ be a closed subset. Let $j : U \to X$ be the inclusion of an open subset with $U \cap Z = \emptyset$. Then $R\mathcal{H}_ Z(Rj_*K) = 0$ for all $K$ in $D(\mathcal{O}_ U)$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}_ U$-modules representing $K$. Then $j_*\mathcal{I}^\bullet$ represents $Rj_*K$. By Lemma 20.32.9 the complex $j_*\mathcal{I}^\bullet$ is a K-injective complex of $\mathcal{O}_ X$-modules. Hence $\mathcal{H}_ Z(j_*\mathcal{I}^\bullet )$ represents $R\mathcal{H}_ Z(Rj_*K)$. Thus it suffices to show that $\mathcal{H}_ Z(j_*\mathcal{G}) = 0$ for any abelian sheaf $\mathcal{G}$ on $U$. Thus we have to show that a section $s$ of $j_*\mathcal{G}$ over some open $W$ which is supported on $W \cap Z$ is zero. The support condition means that $s|_{W \setminus W \cap Z} = 0$. Since $j_*\mathcal{G}(W) = \mathcal{G}(U \cap W) = j_*\mathcal{G}(W \setminus W \cap Z)$ this implies that $s$ is zero as desired. $\square$

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