Timeline for Intuition for Group Cohomology
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 17, 2018 at 12:19 | comment | added | LSpice | I also like the fact that this makes clear that there is nothing necessarily geometric about the parameterisation of rational points on a unit circle; the same approach works just as well to parameterise solutions to $x^2 + y^2 = 1$ over any field in which $-1$ is not a square. | |
| Jun 11, 2017 at 5:30 | comment | added | Noam D. Elkies | It works for any conic with a rational point once you've checked that it can be transformed to the unit circle in a quadratic field extension. But it's not really group cohomology, just the $n=2$ case of Hilbert's actual Theorem 90 (as opposed to $H^1(G,K^*)=0$, which as I understand was itself proved by Emmy Noether for arbitrary finite $G$ long before it was re-proved by using properties of group cohomology to reduce it to Hilbert $-$ and then renamed to make it seem that Hilbert proved the full result, and in its $H^1$ guise at that). | |
| Jan 6, 2013 at 9:38 | comment | added | Mariano Suárez-Álvarez | Is this example one of a kind or just a fluke? | |
| Nov 14, 2012 at 5:32 | comment | added | Franz Lemmermeyer | This example was used by Emil Artin in his Hamburg lectures on algebra in I think 1961. | |
| Jun 8, 2011 at 2:01 | comment | added | SGP | @Elencwajg:This is a very nice observation! I will definitely use it as motivation in my teaching! | |
| Jan 8, 2010 at 4:25 | comment | added | Harry Gindi | See p.302 of Serge Lang's algebra in the section on Galois Cohomology for cyclic groups and a more general but still elementary proof of the full theorem in exercise 4 of chapter 20 on p.826. | |
| Jan 6, 2010 at 16:50 | comment | added | S. Carnahan♦ | @Martin: I don't understand your statement about chronology. It is quite possible to prove Theorem 90 without using a parametrization of the rational circle, let alone many of them. Also, I don't see why the existence of a nice geometric proof means one should stop looking for other arguments and points of view. | |
| Jan 6, 2010 at 16:28 | comment | added | Martin Brandenburg | I don't like this folklore application because before having done the general proof of $H^1(G,K^*)=0$, where you have to come up with special characters whose linear independancy is used, you have already parameterized the rational circle many times; besides, there are nice geometric approaches to it. | |
| Jan 6, 2010 at 12:25 | comment | added | Georges Elencwajg | Oh, I agree that cohomologists must have known this for ages. But Davidac897's question proves that it might be helpful to have elementary examples explicitly written down to show at an early stage that group cohomology can have applications even before the whole somewhat intimidating machinery is developed. | |
| Jan 6, 2010 at 11:31 | comment | added | Qiaochu Yuan | This observation was written down by Elkies, but I can't help but feel like it had to have been known much earlier. | |
| Jan 6, 2010 at 9:46 | history | answered | Georges Elencwajg | CC BY-SA 2.5 |