Timeline for What is an infinite prime in algebraic topology?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 25, 2015 at 21:22 | answer | added | Qiaochu Yuan | timeline score: 10 | |
| Oct 19, 2014 at 3:09 | comment | added | S. Carnahan♦ | I saw a talk by Morava in 2009, where he displayed a picture of the "Berkovich spectrum of the sphere spectrum". All of the finite prime branches had extended bits corresponding to homotopy-theoretic localizations. The archimedean branch ran into a picture of a dragon, labeled "$C^*$-algebras?". | |
| Oct 18, 2014 at 0:15 | answer | added | André Henriques | timeline score: 10 | |
| Oct 12, 2014 at 15:41 | comment | added | Donu Arapura | If you take a $\mathbb{Q}$-algebra, for example a class in the Brauer group or a cohomology ring of a space, and tensor by $\mathbb{R}$ you actually loose information (e.g $Br(\mathbb{R})$ is much simpler than $Br(\mathbb{Q})$). Loss of information is not always a bad thing because the resulting objects may be easier to classify… but I guess you are after something else. | |
| Oct 12, 2014 at 8:33 | comment | added | Anton Fetisov | @DonuArapura, that would give the same information as $\otimes \mathbf Q$. Completions don't matter in AT as much as in algebra. In AT it is just a way to study torsion subgroups, while in algebra it greatly simplifies all equations. | |
| Oct 12, 2014 at 7:26 | comment | added | Donu Arapura | I'm not a topologist, so this may be too naive, but we know from work of Quillen and Sullivan that rational homotopy theory is equivalent to the homotopy theory of a DGL or DGA over $\mathbb{Q}$. We could simply tensor this by $\mathbb{R}$ couldn't we? I know that the "real" in the paper "Real homotopy theory of theory of Kahler manifolds" by Deligne, Griffiths, Morgan, Sullivan refers to this process. | |
| Oct 12, 2014 at 3:16 | comment | added | Jeff Strom | I remember reading in a problem session from a long-ago conference a question asked by Haynes Miller saying something like: "Lots of theorems work for all sufficiently large $p$ --- can we make a theory of the infinite prime?" | |
| Oct 12, 2014 at 0:07 | comment | added | Jonathan Beardsley | I would say the people who could really speak to this are Andrew Salch and Jack Morava, neither of whom are, unfortunately, on MathOverflow. | |
| Oct 12, 2014 at 0:02 | comment | added | Jonathan Beardsley | I haven't heard of anything like this. I suppose in some sense one could try to think of "valuations" as corresponding to Bousfield localizations, or at least, localizations that behave like completion (e.g. localization at HF_p or Morava K-theory). There may be a fruitful analogy to be made here between certain Bousfield localizations and a notion of "valuations" on the sphere spectrum. I guess in some sense topology doesn't SEE the Archimedean place. | |
| Oct 11, 2014 at 21:56 | history | asked | Anton Fetisov | CC BY-SA 3.0 |