The Stacks project

Lemma 20.34.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset.

  1. For $K$ in $D(\mathcal{O}_ X|_ Z)$ we have $i_*K$ in $D_ Z(\mathcal{O}_ X)$.

  2. The functor $i_* : D(\mathcal{O}_ X|_ Z) \to D_ Z(\mathcal{O}_ X)$ is an equivalence with quasi-inverse $i^{-1}|_{D_ Z(\mathcal{O}_ X)} = R\mathcal{H}_ Z|_{D_ Z(\mathcal{O}_ X)}$.

  3. The functor $i_* \circ R\mathcal{H}_ Z : D(\mathcal{O}_ X) \to D_ Z(\mathcal{O}_ X)$ is right adjoint to the inclusion functor $D_ Z(\mathcal{O}_ X) \to D(\mathcal{O}_ X)$.

Proof. Part (1) is immediate from the definitions. Part (3) is a formal consequence of part (2) and Lemma 20.34.1. In the rest of the proof we prove part (2).

Let us think of $i$ as the morphism of ringed spaces $i : (Z, \mathcal{O}_ X|_ Z) \to (X, \mathcal{O}_ X)$. Recall that $i^*$ and $i_*$ is an adjoint pair of functors. Since $i$ is a closed immersion, $i_*$ is exact. Since $i^{-1}\mathcal{O}_ X = \mathcal{O}_ X|_ Z$ is the structure sheaf of $(Z, \mathcal{O}_ X|_ Z)$ we see that $i^* = i^{-1}$ is exact and we see that that $i^*i_* = i^{-1}i_*$ is isomorphic to the identify functor. See Modules, Lemmas 17.3.3 and 17.6.1. Thus $i_* : D(\mathcal{O}_ X|_ Z) \to D_ Z(\mathcal{O}_ X)$ is fully faithful and $i^{-1}$ determines a left inverse. On the other hand, suppose that $K$ is an object of $D_ Z(\mathcal{O}_ X)$ 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 $i_* : D(\mathcal{O}_ X|_ Z) \to D_ Z(\mathcal{O}_ X)$ is an equivalence.

To finish the proof it suffices to show that $R\mathcal{H}_ Z(K) = i^{-1}K$ if $K$ is an object of $D_ Z(\mathcal{O}_ X)$. To do this we can use that $K = i_*i^{-1}K$ as we've just proved this is the case. Then Lemma 20.34.1 tells us what we want. $\square$

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