Timeline for finite etale group scheme over a field
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 5, 2012 at 7:19 | comment | added | Laurent Moret-Bailly | @stefan: every torsor under an étale group scheme is étale as a scheme (or algebraic space), because étaleness is an fppf-local property. Hence it is trivialized by an étale covering (namely itself). | |
| Nov 4, 2012 at 16:25 | comment | added | Chandan Singh Dalawat | If you have a surjective morphism of commutative groups $G\to H$ with kernel $N$, then every fibre is a torsor under $N$. | |
| Nov 4, 2012 at 16:25 | comment | added | Will Sawin | Yes. Let $x$ be an element which is not an $n$th power in $k$. Then the set of $n$th roots of $x$ is a torsor for the $n$th roots of unity in an obvious way, and has no $k$-valued points, thus is nontrivial. | |
| Nov 4, 2012 at 16:22 | comment | added | stefan | the torsor is supposed to be an etale torsor (i.e. trivialized by an etale covering) | |
| Nov 4, 2012 at 16:20 | comment | added | stefan | Thank you very much! Do you have an example of a non-trivial torsor over k with group $\mu_n$ in this situation? | |
| Nov 4, 2012 at 16:08 | comment | added | Chandan Singh Dalawat | For $k=\mathbf{Q}$, it works for every $n>2$. | |
| Nov 4, 2012 at 15:59 | comment | added | stefan | yes, that's right, i was just thinking about this. in fact one could take k to be a finite field, then for large enough n, this should work. Thank you very much. | |
| Nov 4, 2012 at 15:27 | comment | added | Chandan Singh Dalawat | How about $\mu_n$, the $n$-th roots of $1$, over a field $k$ in which $n$ is invertible but which does not contain a primitive $n$-root of $1$ ? | |
| Nov 4, 2012 at 15:19 | history | asked | stefan | CC BY-SA 3.0 |