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I'm thinking about the Grothendieck spectral sequence. I'm trying to understand how it relates to exact couples. Can anyone help me prove in general that the Grothendieck Spectral sequence converges as it should using exact couples?

I'm also trying to relate the definition of an exact couple to the definition of an exact triangle in a triangulated category. It seems like an exact triangle $$N_* \rightarrow M_* \rightarrow L_* \rightarrow N_*[1]$$ Would give an example of an exact couple. Is this true? If so, what is a conceptual understanding of the couple that an exact couple converges to?

I'm also interested in general in a conceptual interpretation of the derived exact couple, especially from the point of view of spectra.

It would be good to build up some intuition as to why exact couples would converge to something of interest, and as to the conceptual significance of the derived exact couple.

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    $\begingroup$ If two of the three objects in your exact triangle coincide, then you can build an exact couple. But if all three are pairwise distinct, there's no clear way that an exact couple gets associated to the exact triangle. $\endgroup$
    – user164898
    Commented Aug 10, 2022 at 21:21
  • $\begingroup$ @A.S. Now, if the triangle is $D \stackrel{\alpha}{\rightarrow} D \rightarrow E \rightarrow D[1]$, then isn't it true that the $n$th page is determined by the fact that one of the elements of the triangle has got to be $D \stackrel{\alpha^n}{\rightarrow} D$? $\endgroup$
    – user30211
    Commented Aug 10, 2022 at 21:31
  • $\begingroup$ @Bob Zinckel But it is not like that. `got to be' does not always work. The first derived couple does not have $D$ itself in it. $\endgroup$
    – Wilberd van der Kallen
    Commented Aug 11, 2022 at 6:59
  • $\begingroup$ @WilberdvanderKallen I realized it's just $\alpha$ and I had the definition wrong... $\endgroup$
    – user30211
    Commented Aug 11, 2022 at 14:33

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