Timeline for Galois module theory: from global to local
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 11, 2021 at 18:25 | vote | accept | Lios | ||
| Jun 11, 2021 at 18:25 | comment | added | Lios | Ok, I see. Again, thank you for all these references and details. | |
| Jun 11, 2021 at 17:42 | comment | added | Henri Johnston | There are also other ways. But it's probably easiest if to wait until Fabio's preprint comes out and then I can explain with reference to the results discussed there. | |
| Jun 11, 2021 at 17:36 | comment | added | Henri Johnston | One way to is to prove this (at least in the case that $p$ is odd) is to use the Grunwald-Wang Theorem; see Cohomology of Number Fields (9.2.8). This pre-dates Henniart's result by at least 5 decades. | |
| Jun 11, 2021 at 17:27 | comment | added | Henri Johnston | In the preface of Child's book he talks states Leopoldt's Theorem for finite abelian extension of $\mathbb{Q}$ and then says "This implies the corresponding result for abelian extensions of $\mathbb{Q}_p$". He does not use the word "immediately" as you claim. There are several ways to see this. | |
| Jun 11, 2021 at 12:54 | comment | added | Lios | If we have the first two, I believe that we need results about realisability to find what we want. Again, there is nothing wrong with this. Just it seems like "to kill a fly with a cannon", and least in my opinion. | |
| Jun 11, 2021 at 12:53 | comment | added | Lios | 4) I will surely read Fabio's preprint as soon as possible. Still, I have already discussed with him about his summary. It seems to me that he presents relations between global freeness and locally freeness (so with semi-completions), and between locally freeness and freeness in the local case, but not bewteen freeness in the global and local case. | |
| Jun 11, 2021 at 12:50 | comment | added | Lios | 3) Ok so with probably just one exception, where there is a proof in the global case, there is also a direct proof of a local case. Also, thank to Henniart's result, now we know that this is always true when $p\ne 2$, and there is no need to prove again directly the case also in the local setting. This is great. | |
| Jun 11, 2021 at 12:45 | comment | added | Lios | For example, in the preface of Childs' book "Taming wild extensions: Hopf algebras and local Galois module theory", published in 2000, Childs says that the global case "immediately implies the corresponding result for abelian extensions of $\mathbb{Q}_p$. | |
| Jun 11, 2021 at 12:43 | comment | added | Lios | 2) There is no problem a priori in the use of Henniart's result (except when $p=2$ clearly), and of course, once you have this result, the proof in the local case is immediate and direct. But this result is quite complicated, it seems to need a big machinery behind, and finally, I am quite sure that this is not the result that other mathematicians had in mind when they used global to deduce local. | |
| Jun 11, 2021 at 12:39 | comment | added | Lios | 1) That is absolutely my bad, in Bergé's paper we can take $A=\mathbb{Z}_p$, so there is no problem there. | |
| Jun 11, 2021 at 12:38 | comment | added | Lios | Thank you very much for your answer. Let me try to clarify my question. | |
| Jun 11, 2021 at 10:09 | history | answered | Henri Johnston | CC BY-SA 4.0 |