Lemma 17.6.2. Let $X$ be a topological space. Let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be a sheaf on $X$. For $U \subset X$ open set

$\Gamma (U, \mathcal{H}_ Z(\mathcal{F})) = \{ s \in \mathcal{F}(U) \mid \text{ the support of }s\text{ is contained in }Z \cap U\}$

Then $\mathcal{H}_ Z(\mathcal{F})$ is an abelian subsheaf of $\mathcal{F}$. It is the largest abelian subsheaf of $\mathcal{F}$ whose support is contained in $Z$. The construction $\mathcal{F} \mapsto \mathcal{H}_ Z(\mathcal{F})$ is functorial in the abelian sheaf $\mathcal{F}$.

Proof. This follows from Lemma 17.5.2. $\square$

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