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
\[ \mathcal{G} \to f_*f^{-1}\mathcal{G} \to \mathcal{G} \]
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
\[ \xymatrix{ V \ar[r]_{j'} \ar[d]_ f & Y \ar[d]^{\overline{f}} \\ U \ar[r]^ j & X } \]
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
\[ j_!f_*f^{-1}\mathcal{G} = \overline{f}_*j'_!f^{-1}\mathcal{G} \]
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. $\square$
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