Class Formation - Tate's Theorem and The Artin Map

Tate's Theorem and The Artin Map

Tate's theorem in group cohomology is as follows. Suppose that A is a module over a finite group G and a is an element of H2(G,A), such that for every subgroup E of G

• H1(E,A) is trivial, and
• H2(E,A) is generated by Res(a) which has order E.

Then cup product with a is an isomorphism

• Hn(G,Z) → Hn+2(G,A).

If we apply the case n=−2 of Tate's theorem to a class formation, we find that there is an isomorphism

• H−2(E/F,Z) → H0(E/F,AF)

for any normal layer E/F. The group H−2(E/F,Z) is just the abelianization of E/F, and the group H0(E/F,AF) is AE modulo the group of norms of AF. In other words we have an explicit description of the abelianization of the Galois group E/F in terms of AE.

Taking the inverse of this isomorphism gives a homomorphism

AE → abelianization of E/F,

and taking the limit over all open subgroups F gives a homomophism

AE → abelianization of E,

called the Artin map. The Artin map is not necessarily surjective, but has dense image. By the existence theorem below its kernel is the connected component of AE (for class field theory), which is trivial for class field theory of non-archimedean local fields and for function fields, but is non-trivial for archimedean local fields and number fields.