Lemma 59.83.7. In Situation 59.83.1 assume $X$ is smooth. Let $j : U \to X$ an open immersion. Let $\ell $ be a prime number. Let $\mathcal{F} = j_!\underline{\mathbf{Z}/\ell \mathbf{Z}}$. Then statements (1) – (8) hold for $\mathcal{F}$.

**Proof.**
Since $X$ is smooth, it is a disjoint union of smooth curves and hence we may assume $X$ is a curve (i.e., irreducible). Then either $U = \emptyset $ and there is nothing to prove or $U \subset X$ is dense. In this case the lemma follows from Lemmas 59.83.2 and 59.83.6.
$\square$

## 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Your email address will not be published.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 (2)

Comment #3204 by Alexander Schmidt on

Comment #3308 by Johan on