Lemma 59.83.9. In Situation 59.83.1 assume $X$ reduced. Let $j : U \to X$ an open immersion with $U$ connected. Let $\ell $ be a prime number. Let $\mathcal{G}$ a finite locally constant sheaf of $\mathbf{F}_\ell $-vector spaces on $U$. Let $\mathcal{F} = j_!\mathcal{G}$. Then statements (1) – (8) hold for $\mathcal{F}$.

**Proof.**
Let $f : V \to U$ be a finite étale morphism of degree prime to $\ell $ as in Lemma 59.66.2. The discussion in Section 59.66 gives maps

whose composition is an isomorphism. Hence it suffices to prove the lemma with $\mathcal{F} = j_!f_*f^{-1}\mathcal{G}$. By Zariski's Main theorem (More on Morphisms, Lemma 37.43.3) we can choose a diagram

with $\overline{f} : Y \to X$ finite and $j'$ an open immersion with dense image. We may replace $Y$ by its reduction (this does not change $V$ as $V$ is reduced being étale over $U$). Since $f$ is finite and $V$ dense in $Y$ we have $V = U \times _ X Y$. By Lemma 59.70.9 we have

By Lemma 59.83.5 it suffices to consider $j'_!f^{-1}\mathcal{G}$. The existence of the filtration given by Lemma 59.66.2, the fact that $j'_!$ is exact, and Lemma 59.83.4 reduces us to the case $\mathcal{F} = j'_!\underline{\mathbf{Z}/\ell \mathbf{Z}}$ which is Lemma 59.83.8. $\square$

## 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Your email address will not be published.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 #4204 by Nicolas Müller on

Comment #4386 by Johan on