TITLE: CFT via Brauer groups vs via ideles
QUESTION [16 upvotes]: I am interested in the relationship between the following two versions of CFT:
Version 1: (Brauer Group Version)
Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map $inv_v:Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z}$ in a (fairly straightforward) cohomological manner. Then the short sequence:
$$1\rightarrow Br(K)\rightarrow \oplus Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z}\rightarrow 1$$ 
where $\oplus Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z}$ is given by $\sum inv_v$, is exact.
Version 2: (Idele Version) 
We can construct a map $$K^{\times}\backslash \mathbb{I}_K/\prod O_v^{\times}  \to Gal(K^{ab}/K),$$
where $\mathbb{I}_K$ are the ideles associated with $K$, by $$(1,...,1,\pi_v,1,...,1)\mapsto Frob_v.$$ Furthermore, this map induces an isomorphism between the profinite completion of $K^{\times}\backslash \mathbb{I}_K/\prod O_v^{\times}$ and $Gal(K^{ab}/K)$.
My question is: How do these two formulations of Class Field Theory relate to one another. Does one imply the other and vice versa? How does one get from one statement to the other? I have never quite been able to square this circle in my mind, even though I've been exposed to CFT for years. Any help would be greatly appreciated.

REPLY [2 votes]: I think that the most powerful and comprehensive link lies in the cohomological machinery of "class formations" introduced by Artin-Tate in their book "Class field theory", chapter 14. This is almost the last chapter, so nobody expects it to be easily understandable.
But let me try to give an idea by considering first the easier case of local CFT: so how do we relate your 2 versions for a local field K ? Version $1$ gives the "invariant" isomorphism $inv_{K}$ between Br(K) and Q/Z. Version $2$ gives the "reciprocity isomorphism" between the profinite completion of K* and Gal($K^{ab}$/K). Accept for one moment that the profinite topology and the normic toplogy on K* are the same (this is a thechnical point in the proof of the so called "existence theorem"). Then version $2$ is equivalent to saying that   , given a finite abelian extension L/K with Galois group G, there is a canonical isomorphism, the "local reciprocity"map, between K/NL* and G (where N denotes the norm from L to K). Here comes cohomology : Br(K) can be viewed as $H^{2}$($K_s$/K, $K_s$*), and in our case this can be shown to be $H^{2}$($K_{nr}$/K, $K_{nr}$*), where $K_{nr}$ is the maximal unramified extension of K, whose Galois group over K is known to be the profinite completion of (Z, +). This gives at once that  $H^{2}$($L$/K, $L$*) of order n = [L:K] and there is an monomorphism $inv_{L/K}$ from to $H^{2}$($L$/K, $L$*), as well as a generator $u_{L/K}$ which is sent to $1/n$ . But general cohomological machinery says that the cup product by $u_{L/K}$ gives an isomorphism from  $H^{-2}$(L/K, Z) to $H^{0}$(L/K, L$^{*}$) (Tate cohomology) , so we are done, because $H^{0}$(L/K, L$^{*}$) is by definition equal to $K*$/N$L*$.
For a number field K, the proof follows exactly the same steps, replacing the multiplicative group $L*$ by the idèle class C$_{L}$ of L. The technical difficulties lie in the computation of the cohomology of C$_{L}$ (whereas the cohomology of the idèles is easy). Finally, notice that L* and C_${L}$ are the two prototypical examples of class formations.    Ref.: for local (resp. global) CFT, see e.g. chapter 6 (resp. 7) of Cassels-Fröhlich's book "Algebraic number theory".
PS: I'm sorry, but something went wrong with my typing.
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