Lemma 20.21.1. Let $i : Z \to X$ be the inclusion of a closed subset. Let $\mathcal{I}$ be an injective abelian sheaf on $X$. Then $\mathcal{H}_ Z(\mathcal{I})$ is an injective abelian sheaf on $Z$.
20.21 Cohomology with support in a closed subset
This section just discusses the bare minimum – the discussion will be continued in Section 20.34.
Let $X$ be a topological space and let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be an abelian sheaf on $X$. We let
\[ \Gamma _ Z(X, \mathcal{F}) = \{ s \in \mathcal{F}(X) \mid \text{Supp}(s) \subset Z\} \]
be the subset of sections whose support is contained in $Z$. The support of a section is defined in Modules, Definition 17.5.1. Modules, Lemma 17.5.2 implies that $\Gamma _ Z(X, \mathcal{F})$ is a subgroup of $\Gamma (X, \mathcal{F})$. The same lemma guarantees that the assignment $\mathcal{F} \mapsto \Gamma _ Z(X, \mathcal{F})$ is a functor in $\mathcal{F}$. This functor is left exact but not exact in general.
Since the category of abelian sheaves has enough injectives (Injectives, Lemma 19.4.1) we we obtain a right derived functor
\[ R\Gamma _ Z(X, -) : D^+(X) \longrightarrow D^+(\textit{Ab}) \]
by Derived Categories, Lemma 13.20.2. The value of $R\Gamma _ Z(X, -)$ on an object $K$ is computed by representing $K$ by a bounded below complex $\mathcal{I}^\bullet $ of injective abelian sheaves and taking $\Gamma _ Z(X, \mathcal{I}^\bullet )$, see Derived Categories, Lemma 13.20.1. The cohomology groups of an abelian sheaf $\mathcal{F}$ with support in $Z$ defined by $H^ q_ Z(X, \mathcal{F}) = R^ q\Gamma _ Z(X, \mathcal{F})$.
Let $\mathcal{I}$ be an injective abelian sheaf on $X$. Let $U = X \setminus Z$. Then the restriction map $\mathcal{I}(X) \to \mathcal{I}(U)$ is surjective (Lemma 20.8.1) with kernel $\Gamma _ Z(X, \mathcal{I})$. It immediately follows that for $K \in D^+(X)$ there is a distinguished triangle
\[ R\Gamma _ Z(X, K) \to R\Gamma (X, K) \to R\Gamma (U, K) \to R\Gamma _ Z(X, K)[1] \]
in $D^+(\textit{Ab})$. As a consequence we obtain a long exact cohomology sequence
\[ \ldots \to H^ i_ Z(X, K) \to H^ i(X, K) \to H^ i(U, K) \to H^{i + 1}_ Z(X, K) \to \ldots \]
for any $K$ in $D^+(X)$.
For an abelian sheaf $\mathcal{F}$ on $X$ we can consider the subsheaf of sections with support in $Z$, denoted $\mathcal{H}_ Z(\mathcal{F})$, defined by the rule
\[ \mathcal{H}_ Z(\mathcal{F})(U) = \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset U \cap Z\} = \Gamma _{Z \cap U}(U, \mathcal{F}|_ U) \]
Using the equivalence of Modules, Lemma 17.6.1 we may view $\mathcal{H}_ Z(\mathcal{F})$ as an abelian sheaf on $Z$, see Modules, Remark 17.6.2. Thus we obtain a functor
\[ \textit{Ab}(X) \longrightarrow \textit{Ab}(Z),\quad \mathcal{F} \longmapsto \mathcal{H}_ Z(\mathcal{F})\text{ viewed as a sheaf on }Z \]
This functor is left exact, but in general not exact. Exactly as above we obtain a right derived functor
\[ R\mathcal{H}_ Z : D^+(X) \longrightarrow D^+(Z) \]
the derived functor. We set $\mathcal{H}^ q_ Z(\mathcal{F}) = R^ q\mathcal{H}_ Z(\mathcal{F})$ so that $\mathcal{H}^0_ Z(\mathcal{F}) = \mathcal{H}_ Z(\mathcal{F})$.
Observe that we have $\Gamma _ Z(X, \mathcal{F}) = \Gamma (Z, \mathcal{H}_ Z(\mathcal{F}))$ for any abelian sheaf $\mathcal{F}$. By Lemma 20.21.1 below the functor $\mathcal{H}_ Z$ transforms injective abelian sheaves into sheaves right acyclic for $\Gamma (Z, -)$. Thus by Derived Categories, Lemma 13.22.2 we obtain a convergent Grothendieck spectral sequence
\[ E_2^{p, q} = H^ p(Z, \mathcal{H}^ q_ Z(K)) \Rightarrow H^{p + q}_ Z(X, K) \]
functorial in $K$ in $D^+(X)$.
Proof. This follows from Homology, Lemma 12.29.1 as $\mathcal{H}_ Z(-)$ is right adjoint to the exact functor $i_*$. See Modules, Lemmas 17.6.1 and 17.6.3. $\square$
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Comment #2123 by Arun Debray on