# Spectral Sequences and Inverse Limits

I recently learned about spectral sequences (which I am still trying to understand). In class we defined a (co)homologically graded exact couple (of $A_p^n$s and $C_p^n$'s with vertical arrows in the $A$ columns) to be a certain grid of exact sequences, which given some convergence assumptions, allows us to calculate the direct limit over $p$ of the $A_p^n$ in terms of $E_\infty (C_p^n)$ (this is a pretty rough sketch). I know inverse limits in general behave worse than direct limits, but in some nice cases, is there something like spectral sequences to compute inverse limits instead? There are lots of interesting things in number theory that are inverse limits is why I am wondering about this.

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I've heard it said that pretty much every spectral sequence essentially comes from some kind of filtration. Direct limits are naturally filtered, but I don't know about inverse limits... – Aaron Mazel-Gee Mar 10 '11 at 1:01
They do come from a filtration. Unrolled exact couples show you where everything lives! – Sean Tilson Mar 10 '11 at 2:17
Inverse limits come from a filtration? – Aaron Mazel-Gee Mar 10 '11 at 3:59

I think that the best thing to do when you want to start learning about spectral sequences is to do computations, see this answer for an outline: Reference for spectral sequences

(I really believe what I said there, it is not a plug.)

I was going to recommend a paper, but I looked at your profile and that drasticly changes my advice. Ask Mike Boardman! Or at least try to trace through the first part of his conditionally convergent spectral sequences! That paper is really flipping cool. It is not easy, but it isn't really hard. Also, you can probably skip the alternative descriptions (the weird big formulas in the first section) of the $E_r$ terms for now.

Seriously, you want to read that paper.

Edit: I feel bad for not explicitly answering your question, but all I would do is look up something in that paper and mess it up trying to copy it down.

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It was an intimidating paper to pick up. I did mean to offend anybody. Better now? – Sean Tilson Mar 10 '11 at 2:54
Yes :) ${}{}{}$ – Mariano Suárez-Alvarez Mar 10 '11 at 2:58
PS: can't you edit my answers? or do you choose to wield your powers in other ways? (giving me the opportunity to correct my error) Interesting side note, the spaces in dollar signs show up in the box up top describing the first bit of your comment. – Sean Tilson Mar 10 '11 at 3:08
Heh. I do not wield powers. At least, that was not the intention! It was just a light-hearted suggestion (the wording would have been rather different if not!) – Mariano Suárez-Alvarez Mar 10 '11 at 3:11
I really flipping appreciate the reference, and I will start reading it. I have no reason to think that the answer to this question is yes, but I would still like to know if there are any applications of spectral sequences to computing inverse limits of things in number theory. There is a very good chance that this is a bad question since I do not have a good sense of the sorts of things SS's do get applied to, but I would like to know anyway. – Vitaly Lorman Mar 10 '11 at 3:34