Timeline for For a morphism f from a regular scheme, should there exist an open subscheme U of the target such that fibre of f at each point of U is regular
Current License: CC BY-SA 3.0
6 events
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| Jun 13, 2014 at 5:38 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
edited body; edited title
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| Jul 16, 2010 at 0:54 | comment | added | BCnrd | Assume $X$ connected, hence irred. Assume $f$ dominant, or else the complement of closure of the image does the trick. Thus, $S$ is irred (and presumably you assume quasi-compact, so noetherian since excellent). By generic flatness, can assume $f$ flat by passing to dense open in $S$. Generic fiber is regular, so if generic point of $S$ is char. 0 then generic fiber is smooth. Then by flatness, $f$ is smooth over dense open (EGA IV$_4$, sec. 17 somewhere), so win without excellence. False without generic char. 0 (Speyer's example). Final equivalence unclear (I guess false); feels useless. | |
| Jul 16, 2010 at 0:45 | vote | accept | Mikhail Bondarko | ||
| Jul 16, 2010 at 0:39 | answer | added | David E Speyer | timeline score: 14 | |
| Jul 16, 2010 at 0:07 | history | edited | Mikhail Bondarko | CC BY-SA 2.5 |
added 155 characters in body; added 14 characters in body
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| Jul 15, 2010 at 23:15 | history | asked | Mikhail Bondarko | CC BY-SA 2.5 |