The Stacks project

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 $D(\underline{\mathbf{Z}}_ X)$.

  1. There is a canonical map $R\Gamma _ Z(X, K) \to R\Gamma _ Z(X, K_{ab})$ which is an isomorphism in $D(\textit{Ab})$.

  2. 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

\[ \Gamma _ Z(X, \mathcal{I}^\bullet ) \to \Gamma _ Z(X, \mathcal{J}^\bullet ) \]

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

\[ \xymatrix{ R\Gamma _ Z(X, K) \ar[r] \ar[d] & R\Gamma (X, K) \ar[r] \ar[d] & R\Gamma (U, K) \ar[d] \\ R\Gamma _ Z(X, K_{ab}) \ar[r] & R\Gamma (X, K_{ab}) \ar[r] & R\Gamma (U, K_{ab}) } \]

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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