Timeline for finite flat morphism of degree two
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 9, 2013 at 9:47 | comment | added | Damian Rössler | Dear user61789, could you come again, please ? (what do you mean by "ZMT" ? I don't quite understand the sentence...). | |
| Jul 9, 2013 at 5:48 | comment | added | user61789 | Dear Damien: The map is given as finite, so what is ZMT telling us that we don't see by more elementary means (say working over open affines in $Y$, as we may do to verify what you claim)? | |
| Jul 8, 2013 at 22:21 | comment | added | Damian Rössler | If $X$ and $Y$ are also regular then any finite generically etale morphism of degree $2$ between them is flat and $X$ is the normalization of $Y$ in the function field of $X$ (by Zariski's main theorem). It thus naturally carries an extension of the generic involution by the construction of the normalization. | |
| Jul 8, 2013 at 13:36 | comment | added | Abhinav Kumar | Good point, Jason :-) | |
| Jul 8, 2013 at 1:39 | comment | added | user61789 | Since it is finite flat of degree 2, Zariski-locally on the base it is ${\rm{Spec}}(A)\rightarrow{\rm{Spec}}(B)$ where 1 is part of a $B$-basis $\{1,a\}$ of $A$. If $T^2-uT+v$ is the characteristic polynomial of $a$-multiplication on $A$ then $B[T]/(T^2-uT+v)\rightarrow A$ via $X\mapsto a$ is an isomorphism. Then $T \mapsto u-T$ is an $A$-automorphism that does the job when $T^2-uT+v$ is separable over Frac($B$). | |
| Jul 8, 2013 at 0:40 | comment | added | Jason Starr | You are implicitly assuming that $f$ is generically etale. In characteristic 2, there are finite flat morphisms of degree 2 that are purely inseparable. | |
| Jul 7, 2013 at 23:48 | comment | added | Abhinav Kumar | Sure. Consider what happens at the level of rings $B \to A$ and attempt to define the involution there. | |
| Jul 7, 2013 at 23:33 | review | First posts | |||
| Jul 8, 2013 at 0:28 | |||||
| Jul 7, 2013 at 23:13 | history | asked | marker | CC BY-SA 3.0 |