The Stacks project

Remark 17.6.2. Let $X$ be a topological space. Let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be an abelian sheaf on $X$. For $U \subset X$ open set, define

\[ \mathcal{H}_ Z(\mathcal{F})(U) = \{ s \in \mathcal{F}(U) \mid \text{ the support of }s\text{ is contained in }Z \cap U\} . \]

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 $Z$. By Lemma 17.6.1 we may (and we do) view $\mathcal{H}_ Z(\mathcal{F})$ as an abelian sheaf on $Z$. In this way we obtain a left exact functor

\[ \textit{Ab}(X) \longrightarrow \textit{Ab}(Z),\quad \mathcal{F} \longmapsto \mathcal{H}_ Z(\mathcal{F}) \text{ viewed as abelian sheaf on }Z \]

All of the statements made above follow directly from Lemma 17.5.2.

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