I am having trouble in understanding Theorem 8.2 of [Adams's book](https://press.uchicago.edu/ucp/books/book/chicago/S/bo21302708.html) and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this case will be


$$\varprojlim\_{a}{}^{p}E^{q}(X\_{a})\implies E^{p+q}(X)$$
where $X\_{a}$ are the skeletons of $X$, $\varprojlim\_{a}{}^{p}$ is the $p-$th right derived limit as defined in the same chapter of Adams.
As far as I understood, we start with a spectral sequence with the first page being $E\_{1}:=E^{p+q}(X\_{p},X\_{p-1})$. So, I have many questions.


1. How do we prove that the second page is given by $\varprojlim\_{a}{}^{p}E^{q}(X\_{a})$?
2. What is the notion of "convergence" here? Is it the condition one he mentions which says $E\_{\infty}^{p,q}\to\varprojlim\_{r}E\_{r}^{p,q}$ is isomorphism? And how are the maps $E\_{r+1}^{p,q}\to E\_{r}^{p,q}$ defined and why being monomorphism implies that the limit exists?
3. What are exactly the filtration quotients of $E^{p+q}(X)$ in condition 3? And how is the exact sequence constructed (the exact sequence in condition 3)?
4. How do we even use theorem 8.2 in Adam's book to provide the exact sequence? This means, how do we verify condition (ii) of this theorem?



Spectral sequence in Adam's book, Theorem 8.2
This is not a complete answer, but it is too long for a comment. I will assume you are interested in the spectral sequences in Section 8 of Adams, not the Atiyah-Hirzebruch spectral sequences of Section 7. There are two different spectral sequences there, depending on whether you start with a CW complex filtered by its skeleta (or any linearly ordered system of subcomplexes) as just before items 1-3:
$$
0 = X\_{-1} \subset X\_0 \subset X\_1 \subset \dots \subset X,
$$
or a CW complex "which is the union of a directed set of subcomplexes $X\_{\alpha}$" (end of the section). For the first of these the higher derived functors vanish. For the second, they need not.


Answers to some of your questions:


1. Not an answer, but a note that when the higher derived functors $\lim^n$ are zero for $n \geq 2$, then a spectral sequence of the form at the end of the section will degenerate to a short exact sequence as in Proposition 8.1. So try to generalize the proof of 8.1 to the situation of an arbitrary directed system.
2. The convergence condition is items 1-3, which he proves (Theorem 8.2) is equivalent to just item 2, and item 2 is typically verified using a Mittag-Leffler condition (exercise just before Prop 8.1).
3. The spectral sequence should converge to $E^{p+q}(X)$, and this is filtered by the groups $E^{p+q}(X\_p)$. That is, we have maps
$$
E^{p+q}(X) \to E^{p+q}(X\_p)
$$
and I think we want to define $F^{p,q}$ by the exact sequence
$$
0 \to F^{p,q} \to E^{p+q}(X) \to E^{p+q}(X\_p).
$$
Then $E^{p,q}\_{\infty}$ will be the kernel of $F^{p,q} \to F^{p-1,q+1}$.
4. Not sure what you mean by "the exact sequence." Adams' Theorem 8.2 might be the same as [Boardman's](https://www.uio.no/studier/emner/matnat/math/MAT9580/v21/dokumenter/conm239-3597.pdf) Theorem 7.4. Take a look.


