Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$ is a filtration of $X$ by its skeletons $X^i$. Now how to get a module filtration of the cochain complex $C^*(X)$ out of this given skeleton filtration, so that we can get the cohomology spectral sequence $(E_r, d_r)$ of the module $C^*(X)$ associated with the (wanted) filtration such that this spectral sequence satisfies:
(i) First quadrant, (ii) $E_1^{p,q}\cong H^{p+q}(X^p,X^{p-1})$ and (iii) collapses at $E_2$.
I have tried defining the filtration of $C^*(X)$ as follows: Let $C_*(X)$ be the singular chain complex of $X$ filtered by $$0 \subset C_*(X^0)\subset C_*(X^1)\subset \cdots\subset C_*(X^p)\subset \cdots \subset C_*(X).$$ Now define the filtration $F^p$ of $C^*(X)$ by $$F^p = \operatorname{Ann}(C_*(X^{p-1})) = \{f\in C^*(X): f(C_*(X^{p-1}))=0\}.$$ The cohomology spectral sequence obtained from this filtration satisfies (i) above, but I cannot show whether it satisfies (ii) and (iii).
Is the above filtration of $C^*(X)$ correct? If yes how to prove part (ii) and (iii), and if it is not correct then how do we define the filtration of $C^*(X)$.
