I am curious if there is a relatively simple explanation of what is the difference between strong convergence and conditional convergence for Spectral Sequences?

(Hopefully a simpler explanation than this info here: https://ncatlab.org/nlab/show/conditional+convergence)

For instance, I am aware if all the differentials are zero after a certain index ($d^r=0$ for all $r\geq r_0$ for some index $r_0$), then the spectral sequence converges. Is this considered strong convergence?

Thanks for any help.

  • 2
    $\begingroup$ The usual reference for these types of things is Boardman's 'Conditionally convergent spectral sequences' - citeseerx.ist.psu.edu/viewdoc/…. The remark after Theorem 7.1 suggests the answer to your question is that such a spectral sequence strongly converges. $\endgroup$ – Drew Heard Nov 16 '17 at 19:07
  • $\begingroup$ I second Drew's recommendation. The paper is complex, but wonderful. The key ingredients are: A spectral sequence should be associated to a good filtration: the intersection/union of the filtration that should be 0 is very much 0 and conditional convergence gives this by definition. You should also be sure that the respective union/intersection of your filtration actually calculates what you are looking for. In practice, checking these conditions is easy. Under these conditions the only thing to check is that $\endgroup$ – Justin Noel Dec 1 '17 at 18:25
  • $\begingroup$ Unless I'm missing something, as long as for each target group H_k, the part of the spectral sequence calculating E_\infty H_k, has no differentials in or out after the nth page (which can depend on k) then the spectral sequence will strongly converge under these conditions. $\endgroup$ – Justin Noel Dec 1 '17 at 18:45

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