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I have begun to read a bit about Spectral Sequences. I have been studying some topics of algebraic topology and this "tool" came up from time to time. I didn't need to go deep, but I felt I had to read a bit. Things started to get more delicate, e.g. the Adams S.S., and more stuff. So I wondered (before attempting this), if these things are important in their own right. I guess so, but I wonder what part of mathematics concentrate on them, and what "gadgets" and conjectures exist about them, independently if may be of where you will use them. I would ask for reference, also, thanks in advance.

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  • $\begingroup$ I tried to add the reference request tag to your post - is there a relevant related field you also wanted to look at? I'm not familiar with Spectral Sequences, but that sounds a bit like Linear Algebra, while you have mentioned it in algebraic topology and tagged category theory ... $\endgroup$ – theREALyumdub Sep 28 '18 at 23:45
  • $\begingroup$ I think a lot of spectral sequences can be viewed as special cases of the spectral sequence where $E^2_{ij} = R^i F (R^j G (X))$ and $\bigoplus_{i+j=n} E^\infty_{ij} \simeq R^n (F \circ G)(X)$ - which is a fairly general category theoretic construction relating compositions of derived functors to derived functors of compositions. (Not completely sure about the details of this, though, so I'm not sure I could expand this to a full answer.) $\endgroup$ – Daniel Schepler Sep 28 '18 at 23:46
  • $\begingroup$ Thanks, interesting viewpoint, I 'll try to check it out $\endgroup$ – vanmeri Sep 28 '18 at 23:51
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Spectral sequences are very useful, but they’re not studied in their own right. The book of McCleary, “User’s Guide to Spectral Sequences”, is the most extensive reference.

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Actually, spectral sequences are studied in their own right as a technical tool. The first spectral sequences you see are probably the bounded ones, like first-quadrant spectral sequences. Those are pretty easy to use. But there are some subtle convergence questions that arose when more advanced spectral sequences were necessary.

A classic is Boardman's "Conditionally Convergent Spectral Sequences", where he develops convergent criteria for half-plane and whole-plane spectral sequences. The main focus of this paper is the study of spectral sequences.

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  • $\begingroup$ Those are good news! $\endgroup$ – vanmeri Oct 4 '18 at 20:09

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