Timeline for G-torsor whose ring of regular functions is a field.
Current License: CC BY-SA 3.0
3 events
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| Aug 23, 2012 at 4:57 | history | edited | Karl Schwede |
fixed a tag
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| Aug 13, 2012 at 0:00 | comment | added | user22479 | No, over any field $k$. The torsor $P$ is affine, and $k[G]$ is exhausted by finitely generated Hopf subalgebras. An injection between Hopf $k$-algebras is faithfully flat, so we get a decreasing system $\{N_i\}$ of closed normal subgroups of $G$ such that the fpqc sheaf $G_i = G/N_i$ is affine of finite type. The quotient sheaf $P_i = P/N_i$ is a $G_i$-torsor, affine of finite type with $P \rightarrow P_i$ faithfully flat and $k[P_i]$'s exhausting $k[P]$. But $P = {\rm{Spec}}(F)$ for a field $F$, so $P_i = {\rm{Spec}}(F_i)$ for fields $F_i$, and $[F_i:k] < \infty$ (!), so $F/k$ is algebraic. | |
| Aug 12, 2012 at 23:43 | history | asked | AFK | CC BY-SA 3.0 |