Timeline for K(r)-localization and monochromatic layers in the chromatic spectral sequence
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 16, 2013 at 2:44 | answer | added | Eric Peterson | timeline score: 20 | |
| Oct 7, 2012 at 21:47 | comment | added | Eric Peterson | @Charles: Ahh, OK, I see some of your point. You can build an $E(R)$-CSS converging to the $E(R)$-Adams SS's $E_2$-page via the same sexseqs as above. This produces the $(r < R)$-truncated CSS, avoids my confusion by keeping $Y = S$, and points out that a crucial component is flatness/Landweber-ness of $E(R)$. You can also get the $(r = R)$-truncated CSS from the $E_R^\vee$-CSS. That's enough to know what I'll say in the notes, but I'm still curious about how or whether $M_R$ belongs in this picture. A $BP$-CSS for $M_r S$ would require some flatness of $BP_* M_r S$ that I doubt we have...? | |
| Oct 7, 2012 at 20:58 | comment | added | Eric Peterson | @Drew: I've opened it many times, but I still haven't properly read it; it's a very dense paper! That spectral sequence probably should have been mentioned above; it would also be nice to know what happens when using $BP_*$ rather than $\pi_*$ on the tower of fibrations $L_{E(r)} \mathbb{S} \to L_{E(r−1)} \mathbb{S}$ and whether the resulting spectral sequence is connected to the CSS. | |
| Oct 7, 2012 at 5:00 | comment | added | Drew Heard | In particular, I hadn't realised that there was a spectral sequence $E_1^{n,k} = \bigoplus_{n=0}^2 \pi_k M_n(S) \Rightarrow \pi_k L_{E(2)} S$ for example | |
| Oct 7, 2012 at 4:57 | comment | added | Drew Heard | Good question, something I've been wondering about. Have you read Behren's "The homotopy groups of the $E(2)$-local sphere at $p > 3$, revisited"? There is a lot of good stuff in there! You might be interested in section 7 where he uses the calculations of $H^*(M_1^1)$ and $H^*(M_0^2)$ to calculate the homotopy of the $K(2)$ local sphere, the $E(2)$ local sphere and the $K(2)/E(2)$ local mod $p$ Moore spectrum. | |
| Oct 7, 2012 at 3:46 | comment | added | Charles Rezk | This deserves a real answer. But in the meantime, let me ask a question: What does "CSS for Y" mean (where Y is some random spectrum)? | |
| Oct 6, 2012 at 20:35 | history | edited | Eric Peterson | CC BY-SA 3.0 |
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| Oct 6, 2012 at 20:29 | history | asked | Eric Peterson | CC BY-SA 3.0 |