Question

This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in chapter IV, proposition 6.2, that his class field axiom implies that the tate cohomology groups H^n(G(L|K),U_L) for n=0,-1 vanish for finite unramified extensions L|K, where U_L is the group of units. He mentions in the proof that every element a \in A_L can be written as a = \epsilon * \pi_K^m, where \epsilon \in U_L and \pi_K is a prime element in A_K. Why does this work? I absolutely understand this argument, when the image of the valuation just lies in \ZZ! But how does this work for a valuation whose image is \widehat{\ZZ}? Unless A is not a profinite module, I don't know what \pi_K^m is for some general m \in \widehat{\ZZ}. Unfortunately this has to work in this generality for global class field theory.

(\ZZ denotes the integers of course, sorry for my personal notation.)

Answer 1

You dont need to make sense out of pi_K^m for a general m in Z-hat. All you really need to know for his argument is that v_K(A_K) = v_L(A_L) as subgroups of Z-hat. I didn't think this through but I think it should be pretty easy to establish from the fact that pi_K is prime for both valuations.

All he really uses is that the Galois group fixes pi_K.

Answer 2

I think Neukirch is actually taking $a$ in $A_K/N_{L|K} A_L$, and since $v_K$ gives an isomorphism from $A_K/N_{L|K} A_L$ to $Z/nZ=\mathbb{Z}/n\mathbb{Z}$ we can then write $a$ in the form $a=\epsilon\pi^{m}_{K}$. However, Neukirch gave an alternative proof for this statement in his book $Class$ $Field$ $Theory$, which doesn't have this problem.