Lemma 59.67.2. Let $\ell$ be a prime number and $n$ an integer $> 0$. Let $K$ be a field with $G = Gal(K^{sep}/K)$ and let $H \subset G$ be a maximal pro-$\ell$ subgroup with $L/K$ being the corresponding field extension. Then $H^ n_{\acute{e}tale}(\mathop{\mathrm{Spec}}(K), \mathcal{F}) = 0$ for all $\ell$-power torsion $\mathcal{F}$ if and only if $H^ n_{\acute{e}tale}(\mathop{\mathrm{Spec}}(L), \underline{\mathbf{Z}/\ell \mathbf{Z}}) = 0$.

Proof. Write $L = \bigcup L_ i$ as the union of its finite subextensions over $K$. Our choice of $H$ implies that $[L_ i : K]$ is prime to $\ell$. Thus $\mathop{\mathrm{Spec}}(L) = \mathop{\mathrm{lim}}\nolimits _{i \in I} \mathop{\mathrm{Spec}}(L_ i)$ as in Lemma 59.67.1. Thus we may replace $K$ by $L$ and assume that the absolute Galois group $G$ of $K$ is a profinite pro-$\ell$ group.

Assume $H^ n(\mathop{\mathrm{Spec}}(K), \underline{\mathbf{Z}/\ell \mathbf{Z}}) = 0$. Let $\mathcal{F}$ be an $\ell$-power torsion sheaf on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$. We will show that $H^ n_{\acute{e}tale}(\mathop{\mathrm{Spec}}(K), \mathcal{F}) = 0$. By the correspondence specified in Lemma 59.59.1 our sheaf $\mathcal{F}$ corresponds to an $\ell$-power torsion $G$-module $M$. Any finite set of elements $x_1, \ldots , x_ m \in M$ must be fixed by an open subgroup $U$ by continuity. Let $M'$ be the module spanned by the orbits of $x_1, \ldots , x_ m$. This is a finite abelian $\ell$-group as each $x_ i$ is killed by a power of $\ell$ and the orbits are finite. Since $M$ is the filtered colimit of these submodules $M'$, we see that $\mathcal{F}$ is the filtered colimit of the corresponding subsheaves $\mathcal{F}' \subset \mathcal{F}$. Applying Theorem 59.51.3 to this colimit, we reduce to the case where $\mathcal{F}$ is a finite locally constant sheaf.

Let $M$ be a finite abelian $\ell$-group with a continuous action of the profinite pro-$\ell$ group $G$. Then there is a $G$-invariant filtration

$0 = M_0 \subset M_1 \subset \ldots \subset M_ r = M$

such that $M_{i + 1}/M_ i \cong \mathbf{Z}/\ell \mathbf{Z}$ with trivial $G$-action (this is a simple lemma on representation theory of finite groups; insert future reference here). Thus the corresponding sheaf $\mathcal{F}$ has a filtration

$0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ r = \mathcal{F}$

with successive quotients isomorphic to $\underline{\mathbf{Z}/\ell \mathbf{Z}}$. Thus by induction and the long exact cohomology sequence we conclude. $\square$

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