Timeline for Non-split extension of the rationals by the integers
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2012 at 7:23 | vote | accept | Mark Opitz | ||
| Aug 23, 2012 at 23:27 | comment | added | Ralph | ... Then I used $f: \mathbb{Q} \xrightarrow{\kappa} \mathbb{Q}/\mathbb{Z} \twoheadrightarrow V_p \hookrightarrow \mathbb{Q}/\mathbb{Z}$. Now $(x,y) \in A$ iff $\bar{x}_p = \bar{y}_p$ and $\bar{y}_r = 0$ if $r \neq p$ where $\kappa(x)=(\bar{x}_r)_r$. But this is equivalent to the description above. | |
| Aug 23, 2012 at 23:24 | comment | added | Ralph | @Mark Opitz: By the paper linked by Damian, the connecting hom. $\delta: Hom(\mathbb{Q},\mathbb{Q}/\mathbb{Z}) \to Ext(\mathbb{Q},\mathbb{Z})$ is surjective. Let $\kappa: \mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ be the canonical hom. If $f \in Hom(\mathbb{Q},\mathbb{Q}/\mathbb{Z})$, then the extension corresponding to $\delta(f)$ is the pull-back of $(\kappa,f)$, i.e. $A=\lbrace (x,y) \in \mathbb{Q} \times \mathbb{Q} \mid f(x)=\kappa(y) \rbrace$. Hence it suffices to find a simple $f$. Let $\mathbb{Q}/\mathbb{Z}=\oplus_r V_r$ be the decomposition in r-torsion subgroups. | |
| Aug 23, 2012 at 20:34 | comment | added | Mark Opitz | Great example. Would me mind telling me how you found it ? | |
| Aug 23, 2012 at 20:17 | comment | added | Ralph | That's good! I searched long for the example, but not long enough to find this presentation :) Noah, thanks for showing me. | |
| Aug 23, 2012 at 20:00 | comment | added | Noah Stein | Very nice example! It's not much of a change, but the exact sequence $0\to\mathbb{Z}\to\mathbb{Z}_{(p)}\oplus\mathbb{Z}[p^{-1}]\to\mathbb{Q}\to 0$ is an example for the same reason and to my eyes it is more "symmetric". The second map is $n\mapsto(n,-n)$ and the third is $(s,t)\mapsto s+t$. | |
| Aug 23, 2012 at 19:35 | comment | added | Mark Grant | Lovely! I think this is what I was going for with my inane comment above. | |
| Aug 23, 2012 at 19:21 | history | answered | Ralph | CC BY-SA 3.0 |