Timeline for homotopy fixed points and fixed points
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 27, 2014 at 14:55 | comment | added | Tyler Lawson | @SeanTilson $THH$ preserves localization: $S^{-1} THH(R) = THH(S^{-1} R)$. As a result, by taking an appropriate (homotopy) limit over a Čech nerve you can patch together the values of $THH$ on open subsets; the fact about localization implies that it's independent of the choice of cover (up to equivalence). | |
| Dec 27, 2014 at 10:55 | history | edited | Max | CC BY-SA 3.0 |
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| Dec 26, 2014 at 23:27 | comment | added | Sean Tilson | How do we build $THH$ of a smooth projective variety? I would think that one would glue together $THH$ of some of the affine covers, but I am not sure exactly how I would do that in the "right" way. For some reason, I am unclear on how excisive THH is for genuinely commutative ring spectra. | |
| Dec 25, 2014 at 14:27 | comment | added | Fernando Muro | Qiaochu of course, did you maybe understand anything different from my comments? | |
| Dec 25, 2014 at 13:22 | vote | accept | Max | ||
| Dec 25, 2014 at 11:34 | comment | added | Qiaochu Yuan | @Fernando: I think that people want THH to be a genuinely $S^1$-equivariant spectrum, which in particular means that it is more than a spectrum equipped with an action of $S^1$ up to homotopy, and in particular it is equipped with genuine (not homotopy) fixed point data. At least Tyler Lawson's answer below seems to only make sense under that assumption. | |
| Dec 24, 2014 at 15:42 | answer | added | Tyler Lawson | timeline score: 5 | |
| Dec 24, 2014 at 13:08 | comment | added | Fernando Muro | what I mean is that the homotopy fixed points functor is the total left derived functor of the plain fixed points functor, hence if you apply both functors to a cofibrant object you obtain weakly equivalent results, whereas you don't if you apply them to a non-cofibrant object. This is not much, really, and won't be very helpful, so I include it as a comment. | |
| Dec 23, 2014 at 21:57 | comment | added | Max | could you say more ? | |
| Dec 23, 2014 at 21:55 | comment | added | Fernando Muro | For 1), the answer depends on the explicit model of $THH(X)$ you take. | |
| Dec 23, 2014 at 21:50 | history | asked | Max | CC BY-SA 3.0 |