Timeline for Is formal smoothness a local property?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 25, 2010 at 2:33 | vote | accept | Harry Gindi | ||
| Apr 24, 2010 at 15:23 | answer | added | Torsten Ekedahl | timeline score: 16 | |
| Apr 24, 2010 at 6:45 | comment | added | Harry Gindi | Yes, there is a gap in the proof of 17.1.6 in EGA because the surprisingly difficult statement about projectivity was not proven until Raynaud-Gruson. The details are available at mathoverflow.net/questions/10731/… . Even though the claim is true, it is very different from the question that I'm asking here, which is an even sharper characterization (i.e. we can check the condition on stalks). | |
| Apr 24, 2010 at 5:53 | comment | added | Torsten Ekedahl | Does anyone understand the proof of the claim (EGA IV 17.1.6) that formal smoothness is local (in the Zariski topology)? The authors refer to 16.5.17 but to me it seems that what they really are referring to is 16.5.18 and there they have explicitly made the assumption that $\Omega^1_{X/Y}$ should be locally finitely presented an assumption which is not part of 17.1.6. Until I understand that proof I am not sure I believe the statement that formal smoothness is local which should be weaker than what you ask. | |
| Apr 24, 2010 at 4:03 | comment | added | Harry Gindi | Very interesting. | |
| Apr 24, 2010 at 2:50 | comment | added | BCnrd | The cotangent complex seems to be natural tool for this, via which it should suffice to show projectivity of $R$-module $\Omega^1_ {R/S}$ can be checked on stalks. In other words, one strategy is to determine criteria for when projectivity of some module (without finiteness hypotheses) can be checked on stalks. That sounds hopeless, but familiarity with the techniques of the unique paper of Raynaud and Gruson would seem to be an essential prerequisite for thinking seriously about this (but I do not recommend it; this is a time-waster just like the "weird" spaces of intro point-set topology). | |
| Apr 23, 2010 at 21:37 | history | asked | Harry Gindi | CC BY-SA 2.5 |