Timeline for Functorial point of view for formal schemes
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 23, 2010 at 13:14 | comment | added | Ricky | Thank you Brian, I'll try to adapt the classical proof to the formal case. | |
| Jul 23, 2010 at 12:48 | comment | added | BCnrd | Ricky, it's true that the link in the answer is more closely related to adic formal schemes, but this is still sweeping the issue under the rug by changing the definitions. I think you'll learn more by following my hint above to adapt the argument for usual schemes directly to the formal scheme case (and then focus on formal spectrum of complete local noetherian rings by proving they're determined by points valued in artin local rings...that will also show you how to come back to the viewpoint in the link given in the above answer). | |
| Jul 23, 2010 at 11:40 | comment | added | Ricky | I've note read that paper yet, but maybe the situation is simpler in the case of adic formal schemes? | |
| Jul 23, 2010 at 11:40 | comment | added | BCnrd | Dear Peter: they're closely related, but to prove deep theorems about formal schemes (algebraization thm, formal GAGA...) the geometric definition is crucial. The situation is analogous to schemes: one can make a categorical definition with functors, but to do geometry (involving concepts like irreducibility, connectedness, dimension, etc.) the ringed-space definition is important. So both viewpoints are needed. A good analogy is completion of a module with respect to an ideal and not only having the inverse system; flatness of completion (in noetherian case) is painful to express otherwise. | |
| Jul 23, 2010 at 11:02 | comment | added | Peter LeFanu Lumsdaine | How big is the difference between these formal schemes and EGA's? Would you say they're really essentially different kinds of objects, or just two formalisms for roughly the same idea that have some technical differences in corner cases? And if the latter, what are the pros and cons of each? Speaking as a category theorist, this functorial definition seems much clearer to me than the EGA definition, so I'd hope/expect that it should work better in practice; but IANAAG, so that's a pretty uninformed opinion and I'd love to know what the experts think! | |
| Jul 23, 2010 at 10:32 | comment | added | BCnrd | This reference defines the question out of existence by changing the def'n of formal schemes into certain functors on rings instead of as EGA defines them. It is akin to redefining schemes as certain functors on rings, which has the effect of bypassing the part of the argument with geometric content. So this shifts the work to proving a presence link between "formal scheme" as defined in this link and as in EGA. (Actually, the definition in the link doesn't quite capture all formal schemes as in EGA; it corresponds to a slightly different concept. ) | |
| Jul 23, 2010 at 10:15 | history | answered | Timo Schürg | CC BY-SA 2.5 |