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
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