Timeline for Schemes over ℤ with a “graded existence over 𝔽₁”
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 9, 2011 at 6:22 | comment | added | Dan Petersen | @Greg: If $X$ is smooth and proper over the integers, this is the main theorem of the paper of van den Bogaart and Edixhoven. | |
| Oct 9, 2011 at 0:17 | comment | added | Greg Kuperberg | So suppose that $X$ is a polynomial-count scheme over $\mathbb{Z}$ with respect to finite-field specializations. Is it necessarily also polynomial count in the valuation Euler characteristic sense for $\mathbb{C}$ and $\mathbb{R}$? If in addition, $X(\mathbb{C})$ and/or $X(\mathbb{R})$ is compact, is the polynomial also the Poincaré-Hilbert polynomial of the cohomology as stated? | |
| Oct 8, 2011 at 22:05 | comment | added | Greg Kuperberg | BTW, I don't know if Soulé's notes were "published" either, but they are in the arXiv. Three cheers for the arXiv; it turned 20 recently. | |
| Oct 8, 2011 at 21:34 | comment | added | Felipe Voloch | The field of one element is more about the number field/function field analogy, which I guess will go wrong eventually but has been very fruitful in number theory and is far from exhausted. | |
| Oct 8, 2011 at 21:22 | comment | added | Greg Kuperberg | My impression (which could be fatuous for all I know) is that --- except for the fact that $\mathbb{C}$ has valuation Euler characteristic 1 --- the field with one element can be viewed as a benevolent exercise in self-deception. In the sense that there isn't really any such field and things will go wrong eventually. But maybe my wording was too strong. | |
| Oct 8, 2011 at 21:02 | history | answered | Felipe Voloch | CC BY-SA 3.0 |