Let $ K $ be a finite extension of a $p$-adic field or a number field, L a finite extension of $K$. The following fact holds: $ \text{Gal}(K^{\text{ab}} / K) \rightarrow \text{Gal}(L^{\text{ab}} / L) $, where the arrow is the Transfer (Verlagerung) map, is injective. I wonder whether this is an arithmetic fact or a fact about absolute Galois group of fields in general?
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I'm not sure how to answer the more philosophical question (it's likely you could encode enough of the axioms to force the purely group-theoretic version of the question to be true, but to ask whether that's what's "really" going on....), but it's certainly not true for all pairs of groups that fit in a similar commutative diagram -- in fact, it's not even true for all such pairs of groups coming from similar questions in algebraic number theory. For example, instead of taking the maximal abelian extension of $K$, take the maximal abelian extension of $K$ which is unramified outside of a set of primes, or split completely at a set of primes, or both -- and you'll pick up a kernel to you transfer map (see Gras, Class Field Theory for some specific calculations of kernels like this). A very relevant related topic worth bringing up is the theorem of Gruenberg-Weiss, which gives an impressively vast generalization of the group-theoretic (and hence the ideal-theoretic) principal ideal theorem entirely in terms of kernels of related transfer maps.
answered Mar 19, 2010 at 13:49
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$\begingroup$ Thanks for the very informative answer. May I ask a further question(with the hope that you are familiar with the class formation approach to class field theory): The approach of class field theory via class formation (for example in Serre's Local Fields) has an implicit assumption that the Verlagerung map is injective. This can be seen as follows. The formation is a system $ (G, \{ G_E \}_{E \in X},A) $ with the intention that $ G_E$ is the Galois groups of the finite extension of some field, and $A$ is a $G$ modules. We also let $ A_E$ be the fixed points of $A$ by $G_E$. $\endgroup$– abcdxyzMar 20, 2010 at 6:02
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$\begingroup$ The intension of defining all these thing eventually is to show that $A_E$ is isomorphic to Gal(E^{ab} / E) ( with further conditions of course). But as we can see this definition make the presumption that $A_F \subseteq A_E $ if $E$ is a finite extension of $F$. This is in fact knowing behind one's head that the Verlagerung map is injective. I believe that class field theory for other fields are out of reach, but it it possible to generalize the definition of class formation to accommodate the further situation where we don't have the injectivity of the Verlagerung map? $\endgroup$– abcdxyzMar 20, 2010 at 6:12
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$\begingroup$ A motivation of the above question: In the case of local or global fields, it seems natural to put the following requirement on the an ideal definition of class formation in order that it is well motivated: If we choose A=lim_{\rightarrow} Gal(E^{ab} /E) where the limit is taken over the direct system given by the Verlagerung map we should be able to show that it forms a class formation). But this is not the case with the current definition, because we do not know a priori that the Verlagerung map is injective. $\endgroup$– abcdxyzMar 20, 2010 at 6:33
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$\begingroup$ To be honest, I've never see the injectivity of the transfer map emphasized in such a foundational role. It's probably equivalent to things that are (to me) much more familiar -- the Neukirch's "class field axiom" seems likely to be equivalent (though I don't have any references nearby) since they're both described cohomologically. In fact, the equivalence might be seen directly from the Hochschild-Serre spectral sequence, though again, this is pure speculation. It is exactly this class field axiom that makes abstract class field theory work so this might be exactly what you're looking for. $\endgroup$ Mar 23, 2010 at 17:52
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$\begingroup$ Thank you for the answer. I will look at these materials. $\endgroup$– abcdxyzMar 24, 2010 at 18:42