Remark 59.69.5. Let k be an algebraically closed field. Let n be an integer prime to the characteristic of k. Recall that
We claim there is a canonical isomorphism
What does this mean? This means there is an element 1_ k in H^1_{\acute{e}tale}(\mathbf{G}_{m, k}, \mu _ n) such that for every morphism \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k) the pullback map on étale cohomology for the map \mathbf{G}_{m, k'} \to \mathbf{G}_{m, k} maps 1_ k to 1_{k'}. (In particular this element is fixed under all automorphisms of k.) To see this, consider the \mu _{n, \mathbf{Z}}-torsor \mathbf{G}_{m, \mathbf{Z}} \to \mathbf{G}_{m, \mathbf{Z}}, x \mapsto x^ n. By the identification of torsors with first cohomology, this pulls back to give our canonical elements 1_ k. Twisting back we see that there are canonical identifications
i.e., these isomorphisms are compatible with respect to maps of algebraically closed fields, in particular with respect to automorphisms of k.
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