Timeline for Fracture Squares of Bousfield Localizations of Spectra
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2019 at 10:24 | comment | added | user20948 | @UrsSchreiber I feel very sorry that I was mistaken. I am now very confused. I wanted to say that, the assumption that $F\wedge L_EY\simeq0$ whenever $F\wedge Y\simeq0$ (change the letter $X$ to $Y$ to avoid ambiguity), this is not assumed. On Bauer's notes (and the nLab page), it is only assumed that $F\wedge L_EX\simeq0$ (for spectra $E,F,X$) and he claimed that the fracture square is a pushout diagram, but it seems to me that in the proof, we also need $F\wedge L_EL_FX\simeq0$ which is not an obvious consequence of $F\wedge L_EX\simeq0$ to me. | |
| Mar 27, 2019 at 16:52 | comment | added | Urs Schreiber | Whoever adds the missing clause to the nLab page will get +5 reputation points as well as the nLabHero-badge: | |
| Mar 27, 2019 at 13:24 | comment | added | Neil Strickland | @FrankScience For the obvious counterexample, just exchange $E$ and $F$ in the arithmetic examle above (so now $E=S/p$ and $F=S\mathbb{Q}$), and take $X=\Sigma^{-1}S(\mathbb{Q}/\mathbb{Z})$. Then $F\wedge X=0$ so $L_EL_FX=L_FX=0$ and $L_EX=S^\wedge_p$ and $L_{E\vee F}X=X$. | |
| Mar 27, 2019 at 13:11 | comment | added | user20948 | @UrsSchreiber I am not an expert and I am not able to work out a counterexample where $F\wedge X\simeq0$, $F\wedge L_EX\not\simeq0$ and the fracture square does not hold. | |
| Mar 26, 2019 at 12:09 | comment | added | Urs Schreiber | @FrankScience thanks for your comment. I would have to remind myself of this to give a decent reaction, but right now I am absorbed with something else. If you feel certain that there is something missing, I'd be grateful if you just went ahead and fixed the nLab page (just hit "edit" at the bottom) and leave a comment on your edit in the respective comment box. | |
| Mar 25, 2019 at 11:57 | comment | added | user20948 | @UrsSchreiber It seems to me that the assumption $F\wedge L_EX=0$ is necessary for the lemma, which is not mentioned in Bauer's notes or on the nLab page. | |
| Jul 25, 2014 at 14:58 | comment | added | Urs Schreiber | A good place to read about these statements is Tilman Bauer's notes math.mit.edu/conferences/talbot/2007/tmfproc/Chapter09/… . There is now also an nLab entry ncatlab.org/nlab/show/fracture+theorem | |
| Mar 13, 2012 at 22:38 | comment | added | Jonathan Beardsley | I think the relevant paper here might be Hovey's paper on the chromatic splitting conjecture? | |
| Mar 13, 2012 at 14:47 | comment | added | Jonathan Beardsley | Ah yes Neil thankyou. I see the point now, where smash is a problem. I guess I'm trying to have some kind of descent property, so what you say may indeed work anyway. Thanks! | |
| Mar 13, 2012 at 14:33 | vote | accept | Jonathan Beardsley | ||
| Mar 13, 2012 at 6:39 | history | answered | Neil Strickland | CC BY-SA 3.0 |