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
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 24, 2010 at 5:28 | vote | accept | Anton Geraschenko | ||
| Apr 22, 2010 at 20:13 | comment | added | Harry Gindi | By the way, I was implying that yes, you can make the substitution that you noted in the finitely presented case, but not in the formal case, so my comment is consistent with Anton's comment. I was remarking that the definition is stated the way it is because it is a special case of the formal case. | |
| Apr 22, 2010 at 3:31 | comment | added | Anton Geraschenko | @unknown: Yes, you get the same notions if you just use Artinian local rings; see EGA IV, section 17.4 (particularly Proposition 17.4.2). This is actually much more local than I first expected these properties to be. It wouldn't be unreasonable to expect that you can only reduce down to the case where you replace "affine scheme" by "spectrum of a local ring". | |
| Apr 21, 2010 at 20:13 | comment | added | Harry Gindi | See my question on formally étale morphisms (and the answer from Gabber that I posted). | |
| Apr 21, 2010 at 20:09 | comment | added | Harry Gindi | unknown(google) this is trivial for formally unramified morphisms, a hard theorem for formally étale morphisms, and an open problem for formally smooth morphisms. This is why we require it for all rings in the finitely presented case, because we want étale (resp. smooth, resp. unramified) morphisms to be finitely presented and formally smooth (resp. étale, resp. unramified). You shouldn't build reductions into the definition when they only hold in special cases. | |
| Apr 21, 2010 at 19:08 | answer | added | Bjorn Poonen | timeline score: 24 | |
| Apr 21, 2010 at 8:50 | answer | added | Angelo | timeline score: 8 | |
| Apr 21, 2010 at 8:42 | answer | added | Bjorn Poonen | timeline score: 10 | |
| Apr 21, 2010 at 8:10 | answer | added | Minhyong Kim | timeline score: 9 | |
| Apr 21, 2010 at 6:16 | comment | added | Qfwfq | Is it the same if you substitute "Artinian local ring" for "affine scheme" in the definitions? I would expect smoothness, unramifiedness, and étaleness to be local properties... | |
| Apr 21, 2010 at 5:38 | history | asked | Anton Geraschenko | CC BY-SA 2.5 |