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

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

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