How to compute and to use spectral sequences ?. I dont know the steps to follow to compute a spectral sequence. Do we need to compute all pages : $ E_r $ and all spacs $ E_r^{pq} $ and differentials $ d_r $ for all $ r $ and $ p,q $ ?
I've red differents books on this subject, but i still without knowing clearly how to proceed to compute spectral sequences.
Thanks in advance for your help.
It does depend on what you want to get out of a spectral sequence. An example of a sort of beginners use of spectral sequences is the following proof of the snake lemma:
Suppose you have a bicomplex $$ \require{AMScd} \begin{CD} A @>>> B @>>> C \\ @VVf V @VVgV @VVhV \\ D @>>> E @>>> F \end{CD} $$
where the rows are short exact sequences. If we look at the spectral sequence that begins by taking the homology in the horizontal direction, the $E^1$ page is
$$ \require{AMScd} \begin{CD} 0 & \qquad &0 & \qquad& 0 \\ @VVV @VVV @VVV \\ 0 & & 0 & & 0 \end{CD} $$
and thus we know the spectral sequence converges to zero. Then, if we look at the spectral sequence that begins by taking homology in the vertical direction, the $E^1$ page is
$$ \require{AMScd} \begin{CD} \ker(f) @>>> \ker(g) @>>> \ker(h) \\ \\ \operatorname{coker}(f) @>>> \operatorname{coker}(g) @>>> \operatorname{coker}(h) \end{CD} $$
Now we can puzzle out what the $E^2$ page has to look like:
$$ \require{AMScd} \begin{CD} 0 &\qquad& 0 &\qquad& G \\ \\ H &\qquad& 0 &\qquad& 0 \end{CD} $$ where there is only one nontrivial map: the one $H \to G$. (I don't know how to typeset this diagram in mathjax)
The reason the $E^2$ page must look like this is because the previous analysis proves everything must converge to zero. In the places I've filled in zeroes, there will never ever be another nontrivial differential involving them, so they must be zero on this page.
Similarly, the $E^3$ page must be all zeroes, and so we conclude the differential $H \cong G$ is actually an isomorphism.
The only thing left is to convince ourselves that the maps in the $E^1$ page are the ones appearing in the snake lemma.
I find that you can get a lot of mileage out of even this simple use. There is a large variety of diagrams one can study with relatively simple diagram chases, but when doing it in an ad-hoc fashion it can be difficult to figure out just what diagram chase to go through. Paging through spectral sequence often tells you what the right chase is.