Lemma 59.67.1. Let $\ell$ be a prime number and $n$ an integer $> 0$. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be the limit of a directed system of $S$-schemes each $X_ i \to S$ being finite étale of constant degree relatively prime to $\ell$. The following are equivalent:

1. there exists an $\ell$-power torsion sheaf $\mathcal{G}$ on $S$ such that $H_{\acute{e}tale}^ n(S, \mathcal{G}) \neq 0$ and

2. there exists an $\ell$-power torsion sheaf $\mathcal{F}$ on $X$ such that $H_{\acute{e}tale}^ n(X, \mathcal{F}) \neq 0$.

In fact, given $\mathcal{G}$ we can take $\mathcal{F} = g^{-1}\mathcal{F}$ and given $\mathcal{F}$ we can take $\mathcal{G} = g_*\mathcal{F}$.

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 $\mathcal{G}_ i = g_ i^{-1}\mathcal{G}$ satisfy the conditions of Theorem 59.51.3 with colimit sheaf given by $g^{-1}\mathcal{G}$. This tells us that:

$\mathop{\mathrm{colim}}\nolimits _{i\in I} H^ n_{\acute{e}tale}(X_ i, g_ i^{-1}\mathcal{G}) = H^ n_{\acute{e}tale}(X, \mathcal{G})$

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

$H^ n_{\acute{e}tale}(S, \mathcal{G}) \to H^ n_{\acute{e}tale}(X_ i, g_ i^{-1}\mathcal{G})$

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 $\mathcal{G} = g_*\mathcal{F}$. $\square$

Comment #5360 by Laurent Moret-Bailly on

In addition to Ben's comment, the converse (constructing $\mathcal{G}$ from $\mathcal{F}$) is proved but not stated. Also, in the proof there are 3 instances of an exponent $p$ standing for $n$.

Comment #5598 by on

Yes, I saw the problem you are pointing out when I fixed Ben's comment. The typos are fixed here. Thanks!

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