Timeline for Does the coordinate ring of affine variety admit a structure of infinite dimensional variety?
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5 events
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| Dec 8, 2013 at 19:48 | comment | added | Qiaochu Yuan | I agree with Scott. If you don't ask for any kind of naturality then this trivially exists, and if you do ask for any kind of naturality then it's unclear if there are any examples of positive dimension (I also don't even know what you would do with this). One thing you could do is weaken a filtration to a filtered colimit and then again this trivially exists, since every vector space is the filtered colimit of its finite-dimensional subspaces, so is naturally an ind-variety. | |
| Dec 7, 2013 at 15:09 | comment | added | meh | If you want to use degree in any way, doesn't that mean that the ideal of the variety should be defined by homogeneous polynomials ? That would allow you to "descend" the degree function. | |
| Dec 7, 2013 at 13:51 | comment | added | Tomasz Lenarcik | In the case of $\mathbb{A}^n$ I was thinking about the filtration given by the degree function. I know it's not canonical in any way, so you're totally right. | |
| Dec 7, 2013 at 13:33 | comment | added | S. Carnahan♦ | If you want the filtration to be part of the structure of the variety, then I don't see why $\mathbb{A}^n$ has a canonical filtration. On the other hand, the coordinate ring of a variety is a countable dimensional $k$-vector space, so a suitable filtration by finite dimensional affine spaces always exists. | |
| Dec 7, 2013 at 12:52 | history | asked | Tomasz Lenarcik | CC BY-SA 3.0 |