Timeline for Is $\mathbb{P}^1$ the only smooth projective curve with a locally split tangent lie algebroid?
Current License: CC BY-SA 3.0
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| Apr 23, 2017 at 13:46 | comment | added | nfdc23 | If $k\ne \overline{k}$ then this recipe to make an etale map to an affine space Zariski-locally around each closed point wouldn't work at closed points whose residue field is not separable over $k$ (e.g., if $p={\rm{char}}(k)>0$ and $C={\rm{Spec}}(k[t])$ then for $a\in k -k^p$ and $P=\{t^p=a\}$, a $k(P)$-basis of $m_P/m_P^2$ is given by $\{t^p-a\}$, for which the associated $k$-map $C\to\mathbf{A}^1_k$ is not etale at $P$). So it is more "relative" to use $f_1,\dots,f_n\in O_X(U)$ for which $\{df_1,\dots,df_n\}$ is a frame of the vector bundle $\Omega^1_{U/k}$, as exist for $U$'s covering $X$. | |
| Apr 23, 2017 at 2:11 | vote | accept | Saal Hardali | ||
| Apr 23, 2017 at 1:08 | history | answered | Will Sawin | CC BY-SA 3.0 |