The Stacks project

Lemma 59.67.3. 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 $H^ q_{\acute{e}tale}(\mathop{\mathrm{Spec}}(K),\mathcal{F}) = 0$ for $q \geq n$ and all $\ell $-torsion sheaves $\mathcal{F}$ if and only if $H^ n_{\acute{e}tale}(\mathop{\mathrm{Spec}}(L), \underline{\mathbf{Z}/\ell \mathbf{Z}}) = 0$.

Proof. The forward direction is trivial, so we need only prove the reverse direction. We proceed by induction on $q$. The case of $q = n$ is Lemma 59.67.2. Now let $\mathcal{F}$ be an $\ell $-power torsion sheaf on $\mathop{\mathrm{Spec}}(K)$. Let $f : \mathop{\mathrm{Spec}}(K^{sep}) \rightarrow \mathop{\mathrm{Spec}}(K)$ be the inclusion of a geometric point. Then consider the exact sequence:

\[ 0 \rightarrow \mathcal{F} \xrightarrow {res} f_* f^{-1} \mathcal{F} \rightarrow f_* f^{-1} \mathcal{F}/\mathcal{F} \rightarrow 0 \]

Note that $K^{sep}$ may be written as the filtered colimit of finite separable extensions. Thus $f$ is the limit of a directed system of finite étale morphisms. We may, as was seen in the proof of Lemma 59.67.1, conclude that $f$ has vanishing higher direct images. Thus, we may express the higher cohomology of $f_* f^{-1} \mathcal{F}$ as the higher cohomology on the geometric point which clearly vanishes. Hence, as everything here is still $\ell $-torsion, we may use the inductive hypothesis in conjunction with the long-exact cohomology sequence to conclude the result for $q + 1$. $\square$

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