# Characterization of weakly convergence of spectral sequences

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.