Let $C$ be a chain complex (in any abelian category) and let $\{F_p\}$ be a decreasing filtration of $C$. It induces a filtration on the homologies of $C$, given by $$F_pH=im(H(F_p)\rightarrow H(C)),$$ and a spectral sequence $E$ such that $$E^0_p\simeq F_pH/F_{p+1}H.$$

We say that $E$ weakly converges to $H(C)$ if for all $n=p+q$ there are integers $r,s$ such that $$0=F_rH_n\subseteq F_{r+1}H_n\subseteq\cdots\subseteq F_{s-1}H_n\subseteq F_sH_n=H_n$$

and ${}^\infty E_{p,q}\simeq F_pH_n/F_{p+1}H_n$.

(Actually, in general, this definition not requires that the filtration must be the filtration I said above, but one can check that the filtration in the convergence coincides with $F_pH=im(H(F_p)\rightarrow H(C))$.

Now, for every $p$, consider $R_p=\cap_r im(H(F_r)\rightarrow H(F_p))$. Note that there exists an canonical map $R_{p+1}\rightarrow R_p$. How to prove that $E$ weakly converges to $H(C)$ if and only if the maps $R_{p+1}\rightarrow R_p$ are injective for all $p$?

I'm following Weibel's book and his hint is: $R_p\subseteq Q_p$ where $Q_p=\varprojlim^1 A^r_p$ where $A^r_p=\{x\in F_p:d(x)\in F_{p+r}\}$. But apart from not getting this inclusion, I can't prove what I want with it. (There are maps $Q_{p+1}\rightarrow Q_p$ such that the equivalence above holds for these ones.)

One the other hand, Cartan & Eilenberg have proven this result in the book "Homological Algebra", but the constrution of the spectral sequence was made in a different way. So I think that "to translate" this prove to my approach is a bigger problem.

I appreciate any advice.


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