I would like to have a question about the re-index convention. Let us consider a spectrum $E$ (I am mainly interested in motivic setting, however let's consider the simplicial case firstly, i.e. $E$ is a $S^1$-spectrum). The homotopy spectral sequence associated to the Postnikov tower of $E$ takes the form

$E^{p,q}_1 = \pi_{-(p+q)}(\Sigma_s^pH\pi_{p}(E)) \Rightarrow \pi_{-(p+q)}(E)$,

where the differential is given by $d_r: E_r^{p,q} \rightarrow E^{p-1,q+r}$. Already here I am not very sure about $d_r$. Some people use $d_r: E^{p,q} \rightarrow E^{p+1,q+r}$. For the above spectral sequence we denote by $\Sigma_s^{p} = S^1_s \wedge \cdots \wedge S^1_s$, the $p$-times suspension of the simplicial circle and $H\pi_p(E)$ is the Eilenberg-Maclane spectrum associated to $\pi_p(E)$.

Now I would like to rewrite this spectral sequence in $E_2$ term by setting $r \mapsto r+1$.My first question is: is the following spectral sequence correct? I am sorry about my stupid question, because I am very confused about the indexes.

$E^{p,q}_2 = H^{p-q}(X,\pi_p(E)) \Rightarrow \pi_{-(p+q)}(E)(X)$, which looks very similarly to the Atiyah-Hirzebruch spectral sequence. (correct me, if this spectral sequence doesn't make sense).

Secondly, how does the differential in this $E_2$-spectral sequence look like? does it still take the form $d_r: E^{p,q}_r \rightarrow E^{p \pm 1,q+r}$?

  • 2
    $\begingroup$ 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 $d_3$. $\endgroup$ – David White Jul 2 '13 at 15:08
  • $\begingroup$ 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 $\endgroup$ – David White Jul 2 '13 at 15:13
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    $\begingroup$ 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). $\endgroup$ – Marc Hoyois Jul 7 '13 at 11:03
  • $\begingroup$ @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 $\endgroup$ – Nikitas Aug 26 '20 at 13:42
  • $\begingroup$ @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 $d_r$. $\endgroup$ – Marc Hoyois Aug 27 '20 at 14:30

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