Timeline for Example of a smooth morphism where you can't lift a map from a nilpotent thickening?
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 21, 2010 at 21:23 | comment | added | Bjorn Poonen | I added an argument for Kevin's general version as a separate answer. | |
| Apr 21, 2010 at 16:08 | comment | added | BCnrd | Kevin, if we pick a point $y \in Y$ such that $X_y$ has positive dimension, we can pass to that fiber to reduce to the case $Y = {\rm{Spec}}(k)$ for a field $k$. Can work locally on $X$, so $X$ affine. The alg. closure of $k$ in $X$ is finite sepble, so can replace $k$ with that so $X$ geom. connected. By Bertini (and Bjorn's version for finite $k$), can then slice $X$ down to a geom. connected smooth affine curve. So Bjorn's case is not as far from the most general case as it may have initially seemed. | |
| Apr 21, 2010 at 9:10 | comment | added | Kevin Buzzard | Bjorn: your example seems to beg the question: if $X\to Y$ is any smooth but not etale morphism, does there always exist $T_0$ and $T$ as above such that the map doesn't lift?! | |
| Apr 21, 2010 at 8:42 | history | answered | Bjorn Poonen | CC BY-SA 2.5 |