Timeline for Model of a scheme regular over the generic point
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Jul 17, 2010 at 0:32 | vote | accept | Mikhail Bondarko | ||
| Jul 16, 2010 at 22:12 | answer | added | Qing Liu | timeline score: 6 | |
| Jul 16, 2010 at 18:04 | history | edited | Mikhail Bondarko | CC BY-SA 2.5 |
added 211 characters in body
|
| Jul 16, 2010 at 18:00 | comment | added | Mikhail Bondarko | I am terribly sorry! I wanted ${X_U}_{red}$ to be smooth (or fibrewise regular) over $U$. I am studying Voevodsky's motives over a base. I started to think that BCnrd's first comment is (more or less) ok for my purposes, since I can start with a variety that is perfect over the perfect closure of $\eta$. I will write an update to my question. | |
| Jul 16, 2010 at 17:54 | history | edited | Mikhail Bondarko | CC BY-SA 2.5 |
added 28 characters in body
|
| Jul 16, 2010 at 17:40 | comment | added | BCnrd | @Q: Oh of course, that was dumb of me to overlook, openness of the regularity locus upstairs (even with the original formulation of the question). I guess it would be best to know what Mikhail is really looking for, since the recent questions all seem to be centered on a theme but the underlying motivation hasn't been stated yet. | |
| Jul 16, 2010 at 17:34 | comment | added | Qing Liu | @Mikhail: Is the question really you want to mean ? Because the existence of a regular model over an open subset $U$ of $S$ is actually immediate, but the former question you refer to is about models with regular fibers. | |
| Jul 16, 2010 at 17:13 | comment | added | BCnrd | @Mikhail: deJong does have a follow-up ("Families of curves and alterations") where he works out a generalization of his alterations result over any excellent integral noetherian base. If the restriction to base a field or dvr in the IHES paper is too restrictive for your needs, maybe his follow-up is sufficient for your needs? | |
| Jul 16, 2010 at 17:02 | comment | added | Mikhail Bondarko | Alterations seem to require some additional restrictions. I thought that my method (extending regular schemes from the generic point) would work over an arbitrary excellent base. Perhaps, such a generalization is not really interesting. | |
| Jul 16, 2010 at 16:56 | comment | added | Qing Liu | Oops, the condition is not necessary. | |
| Jul 16, 2010 at 16:54 | comment | added | Qing Liu | @B.: For the conclusion to be true, the reduced geometric generic fiber $((X_0)_{\bar{k(\eta)}})_{\rm red}$ must be smooth. Your proof shows that this condition is sufficient. I think that just assuming $X_0$ regular is not enough (consider the curve $y^2=x^p-a$ with $a$ not a $p$-th power in characteristic $p>0$, it is even geometrically reduced). | |
| Jul 16, 2010 at 16:51 | comment | added | BCnrd | Whoops!! Mea culpa, the step where I did descent from perfect closure is wrong, since the underlying reduced scheme of the base change to the perfect closure is merely generic smooth, perhaps not smooth if $X_ {\eta}$ wasn't smooth. I'm now a bit doubtful. Is there a reason why alterations aren't enough for your needs? | |
| Jul 16, 2010 at 16:31 | history | edited | BCnrd | CC BY-SA 2.5 |
edited title
|
| Jul 16, 2010 at 16:30 | comment | added | BCnrd | Mikhail, I edited the question (e.g., "generic fiber $S_0$" of $S$ seemed likely to be a generic pt of $S$). The answer to the modified question (which I hope is what you intended) is "yes". By generic flatness can assume $X$ is $S$-flat, and by descent from perfect closure of $k(\eta)$ (over which regular implies smooth) there's a finite purely insep $K/k(\eta)$ such that $((X_0)_K)_ {\rm{red}}$ is $K$-smooth. Spread $K$ to reduced $U$ finite flat radiciel over dense open in $S$. Then $(X_U)_ {\rm{red}}$ is $U$-smooth. Replace $U$ with regular dense open (excellence!), so you win. QED | |
| Jul 16, 2010 at 16:22 | history | edited | BCnrd | CC BY-SA 2.5 |
added 41 characters in body
|
| Jul 16, 2010 at 15:59 | history | asked | Mikhail Bondarko | CC BY-SA 2.5 |