Timeline for Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?
Current License: CC BY-SA 2.5
7 events
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| Jul 22, 2010 at 17:52 | comment | added | David E Speyer | If $R$ is a normal subring of a field $K$, it is common to study $R$ by considering the set of valuations that are nonnegative on $R$. This will distinguish different subrings of $K$, so you can get more information than just the birational class. As far as I know, there is no difficulty extending this construction to non-affine normal schemes, but I don't know a reference for that. | |
| Jul 22, 2010 at 1:05 | comment | added | Tom Goodwillie | In one of the more extreme topologies defined by Voevodsky -- the $h$-topology -- the valuation rings whose fraction fields are algebraically closed play the role of local rings. This observation was a little useful to me once, but I don't know if it has ever been useful to anybody else. | |
| Jul 21, 2010 at 22:38 | comment | added | BCnrd | Bugs, there are more clever/useful ways to define a "Riemann-Zariski space" attached to a scheme, involving structure beyond function fields. Consider Michael Temkin's very creative use of a scheme-like version of Riemann-Zariski spaces in his recent work on semistable curve fibrations and higher-dimensional analogues. So the answer is a definite "yes" to the question of whether spaces of valuations can be useful in non-birational modern algebraic geometry; moreover, it has nothing to do with logic stuff. Grothendieck would spin in his grave over this, except he's still alive (it seems). | |
| Jul 21, 2010 at 21:24 | history | edited | Bugs Bunny | CC BY-SA 2.5 |
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| Jul 21, 2010 at 21:07 | comment | added | Bugs Bunny | Yeah, this is probably right, Doc. | |
| Jul 21, 2010 at 20:45 | comment | added | David Corwin | Fine, then our new object would be like a variety up to birational equivalence. | |
| Jul 21, 2010 at 20:41 | history | answered | Bugs Bunny | CC BY-SA 2.5 |