The Stacks project

Proof. The functor $i_*$ is exact, so $i_* = Ri_* = Li_*$. Hence part (1) of the lemma follows from Modules, Lemma 17.13.6 and Derived Categories, Lemma 13.30.3. Let $K$ be as in (2). We can represent $K$ by a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}_ X|_ Z$-modules. By Lemma 20.32.9 the complex $i_*\mathcal{I}^\bullet$, which represents $i_*K$, is a K-injective complex of $\mathcal{O}_ X$-modules. Thus $R\mathcal{H}_ Z(i_*K)$ is computed by $\mathcal{H}_ Z(i_*\mathcal{I}^\bullet ) = \mathcal{I}^\bullet$ which proves (2). Part (3) is a special case of (2). $\square$

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. $\square$

Proof. Since $i_* : \textit{Mod}(\mathcal{O}_ X|_ Z) \to \textit{Mod}(\mathcal{O}_ X)$ is exact and left adjoint to $\mathcal{H}_ Z$ (Modules, Lemma 17.13.6) this follows from Derived Categories, Lemma 13.31.9. $\square$

Proof. Follows from the construction of right derived functors using K-injective resolutions, Lemma 20.34.3, and the fact that $\Gamma _ Z(X, -) = \Gamma (Z, -) \circ \mathcal{H}_ Z$. $\square$

Proof. Choose a K-injective complex $\mathcal{I}^\bullet$ all of whose terms are injective $\mathcal{O}_ X$-modules representing $K$. See Section 20.28. Recall that $\mathcal{I}^\bullet |_ U$ is a K-injective complex of $\mathcal{O}_ U$-modules, see Lemma 20.32.1. Hence each of the derived functors in the distinguished triangle is gotten by applying the underlying functor to $\mathcal{I}^\bullet$. Hence we find that it suffices to prove that for an injective $\mathcal{O}_ X$-module $\mathcal{I}$ we have a short exact sequence

This follows from Lemma 20.8.1 and the definitions. $\square$

Proof. Choose a K-injective complex $\mathcal{I}^\bullet$ all of whose terms are injective $\mathcal{O}_ X$-modules representing $K$. See Section 20.28. Recall that $\mathcal{I}^\bullet |_ U$ is a K-injective complex of $\mathcal{O}_ U$-modules, see Lemma 20.32.1. Hence each of the derived functors in the distinguished triangle is gotten by applying the underlying functor to $\mathcal{I}^\bullet$. Hence it suffices to prove that for an injective $\mathcal{O}_ X$-module $\mathcal{I}$ we have a short exact sequence

This follows from Lemma 20.8.1 and the definitions. $\square$

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. $\square$

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. $\square$

Proof. Omitted. $\square$

Proof. Omitted. Hints: Using that $H^ p(Z, R\mathcal{H}_ Z(K)) = H^ p(X, i_*R\mathcal{H}_ Z(K))$ and similarly for $R\mathcal{H}_{Z'}(Lf^*K)$ this follows from the functoriality of the pullback maps and the commutative diagram used to define the map of Remark 20.34.12. $\square$