# Spectral sequences on their own right

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.

• 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 ... – theREALyumdub Sep 28 '18 at 23:45
• 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.) – Daniel Schepler Sep 28 '18 at 23:46
• Thanks, interesting viewpoint, I 'll try to check it out – vanmeri Sep 28 '18 at 23:51

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.

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.

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