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

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

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

in $D^+(\textit{Ab})$. As a consequence we obtain a long exact cohomology sequence

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

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

This functor is left exact, but in general not exact. Exactly as above we obtain a right derived functor

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

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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## Comments (1)

Comment #2123 by Arun Debray on