For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global field. I started with the link between étale cohomology and Galois cohomology,
$H^i(\operatorname{Spec}(k)_{ét},F)\cong H^i(G_k,F_k)$
With $F$ a sheaf, $F_k$ its stalk at $\operatorname{Spec}(k^{sep}) \to \operatorname{Spec}(k)$, and $G_k=Gal(k^{sep}/k)$. We can then use the Lyndon-Hochschild-Serre spectral sequence to get the map,
$H^p(G/N,H^q(N,M))\to H^{p+q}(G,M)$
Which then, after making our group $G_k$, our module $F_k$, and our normal subgroup the commutator subgroup of $G_k$, will give us,
$H^p(G_k^{ab},H^q([G_k,G_k],F_k)) \to H^{p+q}(\operatorname{Spec}(k)_{ét},F)$
Then after applying $\hat{C}_k \cong G_k^{ab}$, with $\hat{C}_k$ being the profinite completion of the idele class group, we can finally get,
$H^p(\hat{C}_k,H^q([G_k,G_k],F_k)) \to H^{p+q}(\operatorname{Spec}(k)_{ét},F)$
as our map. Although the connection is weird I was wondering out of interest if there is a simplification, possibly through the $H^q([G_k,G_k],F_k)$ term?
Thanks in advance!
Edit: If I’m not mistaken this map would be a spectral sequence.