The Stacks project

Remark 63.3.9 (Open embeddings and compactly supported sections). Let $X$ be a separated scheme locally of finite type over a field $k$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Exactly as in Remark 63.3.5 for $X' \subset X$ open there is an injective map

\[ H^0_ c(X', \mathcal{F}|_{X'}) \longrightarrow H^0_ c(X, \mathcal{F}) \]

and these maps turn $H^0_ c$ into a “cosheaf” on the Zariski site of $X$.

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