The Stacks project

Proof. Let $g : X \to S$ and $g_ i : X_ i \to S$ denote the structure morphisms. Fix an $\ell $-power torsion sheaf $\mathcal{G}$ on $S$ with $H^ n_{\acute{e}tale}(S, \mathcal{G}) \not= 0$. The system given by $\mathcal{G}_ i = g_ i^{-1}\mathcal{G}$ satisify the conditions of Theorem 59.51.3 with colimit sheaf given by $g^{-1}\mathcal{G}$. This tells us that:

By virtue of the $g_ i$ being finite étale morphism of degree prime to $\ell $ we can apply “la méthode de la trace” and we find the maps

are all injective (and compatible with the transition maps). See Section 59.66. Thus, the colimit is non-zero, i.e., $H^ n(X,g^{-1}\mathcal{G}) \neq 0$, giving us the desired result with $\mathcal{F} = g^{-1}\mathcal{G}$.

Conversely, suppose given an $\ell $-power torsion sheaf $\mathcal{F}$ on $X$ with $H^ n_{\acute{e}tale}(X, \mathcal{F}) \not= 0$. We note that since the $g_ i$ are finite morphisms the higher direct images vanish (Proposition 59.55.2). Then, by applying Lemma 59.51.7 we may also conclude the same for $g$. The vanishing of the higher direct images tells us that $H^ n_{\acute{e}tale}(X, \mathcal{F}) = H^ n(S, g_*\mathcal{F}) \neq 0$ by Leray (Proposition 59.54.2) giving us what we want with $\mathcal{G} = g_*\mathcal{F}$. $\square$