Timeline for some doubts about noncommutative algebraic geometry
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 18, 2011 at 9:06 | comment | added | javier | Indeed, modules have way more information than ideals. IIRC, one of the categorical approaches to NCAG (by Smith, Van den Berg and others) basically relies on considering certain categories that behave in a similar way to categories of modules and just go on as if they were the categories of quasi-coherent sheaves over a noncommutative space. I think that Smith published a paper defining "module points", "module lines" and the like, but cannot find it right now. | |
| Oct 18, 2011 at 4:51 | comment | added | Will Sawin | Weyl has more stuff in it than the commutative limit, since multiple non-abelian rings have the same abelianization. Projective modules over a polynomial ring are free by Quillen-Suslin. Morevover, modules aren't ideals, and don't behave as such. | |
| Oct 17, 2011 at 17:39 | comment | added | Alexander Chervov | Weyl algebra is good example to make me doubt in "... you never have enough of them". Recall this famous result by Hollands (?) Berest Wilson - you classify projective modules over Weyl and you get exactly C^2 - the same amount as in the commutative limit of Weyl e.g. C[x,y]... Moreover you event get Hilbert scheme i.e. glued points... How to be with this ? | |
| Oct 17, 2011 at 14:27 | comment | added | javier | As Zoran says, by "decent" I mean nice enough to allow us to jump back and forth between the ring and the spectrum. In the classical situation if $R$ is a (commutative) ring and $X = Spec(R)$ then one can recover $R$ as the ring of global sections of the structure sheaf. In the noncommutative case, for instance if $R$ is the Weyl algebra $k<x,y>/(yx-xy-1)$ the spectrum is reduced to a single point and the global sections are constant functions. | |
| Oct 17, 2011 at 12:31 | comment | added | Zoran Skoda | He possibly meant faithful enough to reconstruct the scheme back and specializing to commutative one for commutative ring. These are usual minimal requirements. | |
| Oct 17, 2011 at 11:54 | comment | added | Johannes Hahn | @Alexander: "decent" is a synonyme for "nice", not a mathematical notion. | |
| Oct 17, 2011 at 11:23 | comment | added | Alexander Chervov | what means "decent spectrum" ? | |
| Oct 17, 2011 at 10:50 | history | answered | javier | CC BY-SA 3.0 |