Timeline for The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus\{0\}$, the non-zero integers)
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 25, 2023 at 21:12 | comment | added | Emily | Oh, I see! Thank you for explaining it to me :) | |
| May 25, 2023 at 17:27 | comment | added | Tyler Lawson | @Emily Sorry for the late reply. The localization $L_{H\Bbb Z}$ doesn't do anything interesting to connective spectra like the sphere spectrum - they are already homotopy limits of their Postnikov towers, and you can use this to prove that they are local. But it has rather drastic effects on some important nonconnective spectra (the Morava K-theories and E-theories). | |
| Apr 13, 2023 at 21:31 | comment | added | Emily | Oh I had never though about localization of spectra in that way, thanks! Also, this is probably a silly question, but is $L_{H\mathbb{Z}}$ (and in particular $L_{H\mathbb{Z}}\mathbb{S}$) anything interesting? (Also sorry if case the question ends up not making sense; it's been ages since I read about localization, and I haven't had a chance to properly learn it yet) | |
| Apr 13, 2023 at 14:13 | comment | added | Tyler Lawson | @Emily There are other well-known localizations that still "invert things" but those things are no longer strictly represented in the coefficient ring; these are things like K(n) and T(n) localization in chromatic homotopy theory. | |
| Mar 31, 2023 at 22:16 | vote | accept | Emily | ||
| Mar 31, 2023 at 22:16 | comment | added | Emily | Hi Tyler, thank you so much for your answer! I now see why the positive degree elements end up all vanishing. (That said it's such a shame though! I was really hoping there'd be a more interesting ring spectrum counterpart to $\mathbb{Q}$ than just $H\mathbb{Q}$ :/) | |
| Mar 31, 2023 at 17:59 | history | answered | Tyler Lawson | CC BY-SA 4.0 |