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. $\square$

Comment #4204 by Nicolas Müller on

I have a hard time understanding how Lemma 0959 is used here to prove that $j_!f_* f^{-1}\mathcal{G} = \overline{f}_* j'_!f^{-1}\mathcal{G}$. Instead, wouldn't it be better to combine Propositions 03QP and 03S5 to show that both sides agree on the stalks? Then one also doesn't need to argue that the diagram is cartesian. (Also, at least in the preview, the LaTeX doesn't render properly in this comment.)

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