The Stacks project

Lemma 62.12.4. Let $k$ be an algebraically closed field. Let $X$ be a separated scheme of finite type type over $k$ of dimension $\leq 1$. Let $\Lambda $ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda $-modules on $X$ which is torsion. Then $H^ q_ c(X, \mathcal{F})$ is a finite $\Lambda $-module.

Proof. This is a consequence of √Čtale Cohomology, Theorem 59.84.7. Namely, choose a compactification $j : X \to \overline{X}$. After replacing $\overline{X}$ by the scheme theoretic closure of $X$, we see that we may assume $\dim (\overline{X}) \leq 1$. Then $H^ q_ c(X, \mathcal{F}) = H^ q(\overline{X}, j_!\mathcal{F})$ and the theorem applies. $\square$


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