Lemma 20.34.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $Z \subset X$ be a closed subset. Let $K$ be an object of $D(\mathcal{O}_ X)$ and denote $K_{ab}$ its image in $D(\underline{\mathbf{Z}}_ X)$.
There is a canonical map $R\Gamma _ Z(X, K) \to R\Gamma _ Z(X, K_{ab})$ which is an isomorphism in $D(\textit{Ab})$.
There is a canonical map $R\mathcal{H}_ Z(K) \to R\mathcal{H}_ Z(K_{ab})$ which is an isomorphism in $D(\underline{\mathbf{Z}}_ Z)$.
Proof. Proof of (1). The map is constructed as follows. Choose a K-injective complex of $\mathcal{O}_ X$-modules $\mathcal{I}^\bullet $ representing $K$. Choose a quasi-isomorpism $\mathcal{I}^\bullet \to \mathcal{J}^\bullet $ where $\mathcal{J}^\bullet $ is a K-injective complex of abelian groups. Then the map in (1) is given by
determined by the fact that $\Gamma _ Z$ is a functor on abelian sheaves. An easy check shows that the resulting map combined with the canonical maps of Lemma 20.32.7 fit into a morphism of distinguished triangles
of Lemma 20.34.5. Since two of the three arrows are isomorphisms by the lemma cited, we conclude by Derived Categories, Lemma 13.4.3.
The proof of (2) is omitted. Hint: use the same argument with Lemma 20.34.6 for the distinguished triangle. $\square$
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