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 Étale 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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)