Class Formation - The Brauer Group

The Brauer Group

The Brauer groups H2(E/*) of a class formation are defined to be the direct limit of the groups H2(E/F) as F runs over all open subgroups of E. An easy consequence of the vanishing of H1 for all layers is that the groups H2(E/F) are all subgroups of the Brauer group. In local class field theory the Brauer groups are the same as Brauer groups of fields, but in global class field theory the Brauer group of the formation is not the Brauer group of the corresponding global field (though they are related).

The next step is to prove that H2(E/F) is cyclic of order exactly |E/F|; the previous section shows that it has at most this order, so it is sufficient to find some element of order |E/F| in H2(E/F).

For cyclic extensions this is already known. The proof for arbitrary extensions uses a homomorphism from the group G onto the profinite completion of the integers, or in other words a compatible sequence of homomorphisms of G onto the cyclic groups of order n for all n. These homomorphisms are constructed using cyclic cyclotomic extensions. This idea was first used by Chebotarev in his proof of Chebotarev's density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem.

The proof of the existence of an element of order |E/F| for an arbitrary layer proceeds by first constructing a suitable auxiliary cyclic extension of degree |E/F| as above; as this is cyclic, there is an element of order |E/F| in its second cohomology, and this element turns out to be essentially an element of H2(E/F).

This shows that the second cohomology group H2(E/F) of any layer is cyclic of order |E/F|, which completes the verification of the axioms of a class formation. With a little more care in the proofs, we get a canonical generator of H2(E/F), called the fundamental class.

It follows from this that the Brauer group H2(E/*) is (canonically) isomorphic to the group Q/Z, except in the case of the archimedean local fields R and C when it has order 2 or 1.