The Stacks project

Lemma 20.21.3. Let $i : Z \to X$ be the inclusion of a closed subset of a topological space $X$. The map $Ri_* = i_* : D(Z) \to D(X)$ induces an equivalence $D(Z) \to D_ Z(X)$ with quasi-inverse

\[ i^{-1}|_{D_ Z(X)} = R\mathcal{H}_ Z|_{D_ Z(X)} \]

Proof. Recall that $i^{-1}$ and $i_*$ is an adjoint pair of exact functors such that $i^{-1}i_*$ is isomorphic to the identify functor on abelian sheaves. See Modules, Lemmas 17.3.3 and 17.6.1. Thus $i_* : D(Z) \to D_ Z(X)$ is fully faithful and $i^{-1}$ determines a left inverse. On the other hand, suppose that $K$ is an object of $D_ Z(X)$ and consider the adjunction map $K \to i_*i^{-1}K$. Using exactness of $i_*$ and $i^{-1}$ this induces the adjunction maps $H^ n(K) \to i_*i^{-1}H^ n(K)$ on cohomology sheaves. Since these cohomology sheaves are supported on $Z$ we see these adjunction maps are isomorphisms and we conclude that $D(Z) \to D_ Z(X)$ is an equivalence.

To finish the proof we have to show that $R\mathcal{H}_ Z(K) = i^{-1}K$ if $K$ is an object of $D_ Z(X)$. To do this we can use that $K = i_*i^{-1}K$ as we've just proved this is the case. Then we can choose a K-injective representative $\mathcal{I}^\bullet $ for $i^{-1}K$. Since $i_*$ is the right adjoint to the exact functor $i^{-1}$, the complex $i_*\mathcal{I}^\bullet $ is K-injective (Derived Categories, Lemma 13.31.9). We see that $R\mathcal{H}_ Z(K)$ is computed by $\mathcal{H}_ Z(i_*\mathcal{I}^\bullet ) = \mathcal{I}^\bullet $ as desired. $\square$


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