Timeline for Motivic homotopy spectral sequence
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 11, 2020 at 12:26 | comment | added | Nikitas Nikandros | @MarcHoyois Ok thanks ! | |
| Aug 27, 2020 at 14:30 | comment | added | Marc Hoyois | @Nikitas I'm not sure what kind of details you're after. With the notation from my previous comment, the first differential comes from the composite $\Sigma^p \mathrm{cofib}(f_q)\to \Sigma^{p+1} F_{q+1} \to \Sigma^{p+1}\mathrm{cofib}(f_{q+1})$, which explains the $d_1$ indices. Most references probably make different choices in the definition of $E_1^{p,q}$ (and the filtration could go the other way), which leads to different gradings for $d_r$. | |
| Aug 26, 2020 at 13:42 | comment | added | Nikitas Nikandros | @MarcHoyois Could you explain that in more detail ? In all the references of the spectral sequence for a filtered object in a stable homotopy theory only mention the standard (co)homological indexing | |
| Jul 7, 2013 at 11:03 | comment | added | Marc Hoyois | Going from $(p,q)$ to $(p+1,q+r)$ is actually a fairly natural indexing when your spectral sequence comes from a filtered object in a triangulated category (as in the OP's case): if your filtered object is $f_q: F_{q+1}\to F_q$ for $q\in\mathbb{Z}$, it's the indexing you get by setting $E_1^{p,q}=H(\Sigma^p \operatorname{cofib}(f_q))$ where $H$ is some homological functor (eg. $[X,-]$ for some object $X$ or $\pi_0$ if you have a $t$-structure). | |
| Jul 2, 2013 at 15:13 | comment | added | David White | Others have asked spectral sequence questions here many times. I recommend: mathoverflow.net/questions/45036, mathoverflow.net/questions/8052, mathoverflow.net/questions/92700. For double complexes and triply graded things (ie for motivic): mathoverflow.net/questions/93621, mathoverflow.net/questions/110812, mathoverflow.net/questions/86947 | |
| Jul 2, 2013 at 15:08 | comment | added | David White | Indexing conventions are notoriously confusing. I'm no expert, but I've never seen anyone going from (p,q) to (p+1,q+r). Normally it's up and to the left or down and to the right, depending on homology vs cohomology. There's a question of whether $d_r$ should go left/right by 1 or by r. That's the difference between the Adams Indexing Convention and the classical one. A good reference is Mosher and Tangora's book. My other advice for a beginner would be to try and isolate the differentials, so on the $E_2$ page maybe just focus on $d_2$, then go to $E_3$ and $d_3$. | |
| Jul 2, 2013 at 15:03 | review | First posts | |||
| Jul 2, 2013 at 16:32 | |||||
| Jul 2, 2013 at 14:44 | history | asked | Thi | CC BY-SA 3.0 |