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Proof. For $q > 1$ we know that $\xi $ dies in $H^ q(X_{\overline{K}}, (X_{\overline{K}} \to \mathop{\mathrm{Spec}}(K))^{-1}\mathcal{F})$ (Theorem 59.83.10). By Lemma 59.51.5 we see that this means there is a finite extension $K'/K$ such that $\xi $ dies in $H^ q(X_{K'}, (X_{K'} \to \mathop{\mathrm{Spec}}(K))^{-1}\mathcal{F})$. Thus we can take $k' = k$ and $Z = X$ in this case.

Assume $q = 1$. Recall that $\mathcal{F}$ corresponds to a discrete module $M$ with continuous $\text{Gal}_ K$-action, see Lemma 59.59.1. Since $M$ is $n$-torsion, it is the union of finite $\text{Gal}_ K$-stable subgroups. Thus we reduce to the case where $M$ is a finite abelian group annihilated by $n$, see Lemma 59.51.4. After replacing $K$ by a finite extension we may assume that the action of $\text{Gal}_ K$ on $M$ is trivial. Thus we may assume $\mathcal{F} = \underline{M}$ is the constant sheaf with value a finite abelian group $M$ annihilated by $n$.

We can write $M$ as a direct sum of cyclic groups. Any two finite étale Galois coverings whose Galois groups have order invertible in $k$, can be dominated by a third one whose Galois group has order invertible in $k$ (Fundamental Groups, Section 58.7). Thus it suffices to prove the lemma when $M = \mathbf{Z}/d\mathbf{Z}$ where $d | n$.

Assume $M = \mathbf{Z}/d\mathbf{Z}$ where $d | n$. In this case $\overline{\xi } = \xi |_{X_{\overline{K}}}$ is an element of

See Theorem 59.83.10. This group classifies $\mathbf{Z}/d\mathbf{Z}$-torsors, see Cohomology on Sites, Lemma 21.4.3. The torsor corresponding to $\overline{\xi }$ (viewed as a sheaf on $X_{\overline{k}, {\acute{e}tale}}$) in turn gives rise to a finite étale morphism $T \to X_{\overline{k}}$ endowed an action of $\mathbf{Z}/d\mathbf{Z}$ transitive on the fibre of $T$ over $x_{\overline{k}}$, see Lemma 59.64.4. Choose a connected component $T' \subset T$ (if $\overline{\xi }$ has order $d$, then $T$ is already connected). Then $T' \to X_{\overline{k}}$ is a finite étale Galois cover whose Galois group is a subgroup $G \subset \mathbf{Z}/d\mathbf{Z}$ (small detail omitted). Moreover the element $\overline{\xi }$ maps to zero under the map $H^1(X_{\overline{k}}, \mathbf{Z}/d\mathbf{Z}) \to H^1(T', \mathbf{Z}/d\mathbf{Z})$ as this is one of the defining properties of $T$.

Next, we use a limit argument to choose a finite extension $k'/k$ contained in $\overline{k}$ such that $T' \to X_{\overline{k}}$ descends to a finite étale Galois cover $Z \to X_{k'}$ with group $G$. See Limits, Lemmas 32.10.1, 32.8.3, and 32.8.10. After increasing $k'$ we may assume that $Z$ splits over $x_{k'}$. The image of $\xi $ in $H^1(Z_{\overline{K}}, \mathbf{Z}/d\mathbf{Z})$ is zero by construction. Thus by Lemma 59.51.5 we can find a finite subextension $\overline{K}/K'/K$ containing $k'$ such that $\xi $ dies in $H^1(Z_{K'}, \mathbf{Z}/d\mathbf{Z})$ and this finishes the proof. $\square$