Timeline for Injectivity of Transfer (Verlagerung) map
Current License: CC BY-SA 2.5
8 events
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| Mar 24, 2010 at 18:42 | comment | added | abcdxyz | Thank you for the answer. I will look at these materials. | |
| Mar 23, 2010 at 17:52 | comment | added | Cam McLeman | 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. | |
| Mar 20, 2010 at 6:33 | comment | added | abcdxyz | 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. | |
| Mar 20, 2010 at 6:12 | comment | added | abcdxyz | 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? | |
| Mar 20, 2010 at 6:02 | comment | added | abcdxyz | 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 $G_E$. | |
| Mar 20, 2010 at 5:36 | vote | accept | abcdxyz | ||
| Mar 19, 2010 at 14:39 | history | edited | Cam McLeman | CC BY-SA 2.5 |
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| Mar 19, 2010 at 13:49 | history | answered | Cam McLeman | CC BY-SA 2.5 |