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The Stacks project

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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