The Stacks project

Lemma 62.12.3. Let $X$ be a separated scheme of finite type over a field $k$. If $\mathcal{F}$ is a torsion abelian sheaf, then the abelian group $H^0_ c(X, \mathcal{F})$ defined in Definition 62.3.7 agrees with the abelian group $H^0_ c(X, \mathcal{F})$ defined in Definition 62.12.1.

Proof. Choose a compactification $j : X \to \overline{X}$ over $k$. In both cases the group is defined as $H^0(\overline{X}, j_!\mathcal{F})$. This is true for the first version by Lemma 62.3.10 and for the second version by construction. $\square$

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