Timeline for Why are divisible abelian groups important?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Aug 9, 2010 at 4:17 | comment | added | Beren Sanders | As Steve has mentioned, divisible modules and injective modules are the same over a PID. More generally, this is true over any Dedekind domain. | |
| Aug 9, 2010 at 3:06 | answer | added | Emerton | timeline score: 14 | |
| Aug 8, 2010 at 12:51 | comment | added | Keenan Kidwell | <<Continued>> The unique divisibility of actually implies that $H^n(G,\mathbb{Q})=0$ for all finite groups $G$ and $n\geq 1$, since the latter group is killed by $\vert G\vert$, but the unique divisibility of $\mathbb{Q}$ implies that multiplication by $\vert G\vert$ is an isomorphism on cohomology. Taking direct limits one gets the same result for profinite $G$. This is crucial because upon taking the long exact cohomology sequence one gets $\mathbb{Q}/\mathbb{Z}\simeq Hom(\hat{\mathbb{Z}},\mathbb{Q}/\mathbb{Z})\simeq H^2(\hat{\mathbb{Z}},\mathbb{Z})$, which is a part of the invariant map. | |
| Aug 8, 2010 at 12:43 | comment | added | Keenan Kidwell | An example where divisibility is used crucially that came to mind first for me is in constructing the invariant map $Br(K)\simeq\mathbb{Q}/\mathbb{Z}$, where $Br(K)=H^2(K,\mathbb{G}_m)$ is the Brauer group of the non-Archimedean local field $K$. One uses the short exact sequence $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}$ of trivial $G_K$-modules, and the fact that $\mathbb{Q}$ is uniquely divisible implies that its $\geq 1$-dimensional cohomology vanishes. | |
| Aug 8, 2010 at 12:24 | answer | added | David Corwin | timeline score: 4 | |
| Aug 8, 2010 at 4:47 | history | edited | Victor Protsak | CC BY-SA 2.5 |
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| Aug 8, 2010 at 4:37 | answer | added | Victor Protsak | timeline score: 10 | |
| Aug 8, 2010 at 4:12 | history | made wiki | Post Made Community Wiki by Sergei Tropanets | ||
| Aug 8, 2010 at 4:04 | comment | added | Steve D | Over any PID, divisible and injective are the same thing. I think over $\mathbb{Z}$, things are nice because all divisible groups (and therefore all injective $\mathbb{Z}$-modules) are known up to isomorphism. There are lots of books which talk about this. Personally, I would recommend (if you can get your hands on it) Kaplansky's "Infinite Abelian Groups". Besides having a ton of info in about 100 pgs, I think the way he presents divisible groups is very slick and easy-to-follow. | |
| Aug 8, 2010 at 3:45 | history | asked | Sergei Tropanets | CC BY-SA 2.5 |