Timeline for Non-commutative algebraic geometry
Current License: CC BY-SA 4.0
9 events
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| Jun 14, 2022 at 9:15 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
minor typos
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| Dec 15, 2009 at 0:40 | history | edited | Shizhuo Zhang | CC BY-SA 2.5 |
added 348 characters in body
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| Dec 15, 2009 at 0:01 | comment | added | Shizhuo Zhang | And this is enough in general case(say locally affine space,not necessarily scheme). I just learned this from course given by Rosenberg this semester. A good reference is Kontsevich-Rosenberg "Noncommutative space and flat descent" | |
| Dec 14, 2009 at 23:58 | comment | added | Shizhuo Zhang | BUT, from categorical view point, we can define another topology which is sufficient. We can take a family of localization functor whose correspondence multiplicative system is finitely generated. Of course, if we restricted ourselves to commutative affine scheme case, it gives the same topology as Zariski topology. We obtain the principle affine open sets. In general, we can take covers,elements of this cover is morphism u whose inverse image functor is exact and locally finite presentable. If u is localization, then the correspondent multiplicative system if f.g. | |
| Dec 14, 2009 at 23:51 | comment | added | Shizhuo Zhang | As javier mentioned,yes, we do not have good notion of spectrum of ring forms by two-sided ideal because there are not many. For example, Weyl algebra A. if we define specA consists of two-sided ideal,we get trivial topology. Actually we have purely categorical version of this spectrum of A-mod. It is called Coarse Zariski topology.The remark is Coarse Zariski topology is too coarse.(we obtain this definition just go directly from comm to noncomm,so it is not right!) | |
| Dec 14, 2009 at 22:10 | comment | added | javier | Kevin: Ok, then the right reference to point you out to, rather than the book by Van Oystaeyen, is this entry at nLab: ncatlab.org/nlab/show/noncommutative+algebraic+geometry | |
| Dec 14, 2009 at 20:54 | comment | added | Kevin H. Lin | javier: Ok, I suppose that you are right, if you interpret my question literally. But I am still very interested in hearing about any reasonable notions of non-commutative algebraic geometry whatsoever. I guess I should have said that to begin with. | |
| Dec 14, 2009 at 11:59 | comment | added | javier | Well, the lack of nice properties of the category of nc rings is a big deal if you try to stick to the classical definitions (which was the original question in the thread). For instance, to have coproducts/fiber products you need to use free/amalgamated products instead of tensor products, and that destroys properties like Noetherianity, finite generation, and many others. And one big inconvenience of this is that you cannot recover the classical case from the noncommutative one. There are workarounds, of course, but they are not what the thread was about. | |
| Dec 14, 2009 at 4:03 | history | answered | Shizhuo Zhang | CC BY-SA 2.5 |