The Stacks project

Definition 63.3.7. Let $X$ be a separated scheme locally of finite type over a field $k$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. We let $H^0_ c(X, \mathcal{F}) \subset H^0(X, \mathcal{F})$ be the set of sections whose support is proper over $k$. Elements of $H^0_ c(X, \mathcal{F})$ are called sections with compact support.