Weibel's H-book, Milnor's exact sequence for spectral sequence of filtered complex, Theorem 5.5.5

Question

This is a question which I asked on StackExchange first, but might be more suited here.

I got stuck on the proof of Theorem 5.5.5 in Weibel's book. Not only that, but I also could not even find the statement of said theorem in any other source, so I am completely at a loss how to proceed.

The result is called the Eilenberg-Moore Filtration Sequence for complete complexes. Interestingly, I was unable to find this result under this name anywhere else.

Suppose the filtered chain complex $C$ is complete with respect to it's filtration $F_*C$, i.e $C=\lim\limits_{\leftarrow} C/F_pC.$ In this case the exact sequence $$0\longrightarrow\mathbin{\lim\limits_{\longleftarrow}}^1H_{n+1}(C/F_pC)\longrightarrow H_n(C)\longrightarrow\mathbin{\lim\limits_{\longleftarrow}} H_n(C/F_pC)\longrightarrow0$$ takes the form $$0\longrightarrow\bigcap F_pH_n(C)\longrightarrow H_n(C)\longrightarrow H_n(C)/\bigcap F_pH_n(C)\longrightarrow0$$ and $${\lim\limits_{\longleftarrow}} H_n(C/F_pC)\cong {\lim\limits_{\longleftarrow}} H_n(C)/F_pH_n(C). $$

The first exact sequence is Milnor's $\mathbin{\lim\limits_{\longleftarrow}}^1$ exact sequence (Theorem 3.5.8):

Let $C_i$ be a tower of chain complexes satisfying the Mittag-Leffler condition, and set $C=\mathbin{\lim\limits_{\longleftarrow}} C_i$. Then there is an exact sequence for each n: $$0\longrightarrow\mathbin{\lim\limits_{\longleftarrow}}^1H_{n+1}(C_i)\longrightarrow H_n(C)\longrightarrow\mathbin{\lim\limits_{\longleftarrow}} H_n(C_i)\longrightarrow0$$

The proof then proceeds as follows:

Taking the limit of the exact sequence of towers $$0\longrightarrow F_pH_*(C)\longrightarrow H_*(C)\longrightarrow H_*(C)/F_pH_*(C)\longrightarrow0$$ we find that there is a monomorphism $H_*(C)/\bigcap F_pH_*(C)\to {\lim\limits_{\longleftarrow}} H_*(C)/F_pH_*(C)$. Also, there is an exact sequnce $$0\longrightarrow H_*(C)/F_pH_*(C)\to H_*(C/F_pC),$$ which also gives a monomorphism ${\lim\limits_{\longleftarrow}} H_*(C)/F_pH_*(C)\to \lim\limits_{\longleftarrow} H_*(C/F_pC)$. This combines into a monomorphism $H_*(C)/\bigcap F_pH_*(C)\to\lim\limits_{\longleftarrow} H_*(C/F_pC)$.

Then Weibel says to "combine this with the $\mathbin{\lim\limits_{\longleftarrow}}^1$ sequence of 3.5.8". And I don't get this last step at all. It seems we have not yet connected $\mathbin{\lim\limits_{\longleftarrow}}^1$ and $\bigcap F_pH_n(C)$.

This result is used quite a bit in later sections, so I would really like to understand it. I would be thankful for any help here.

Answer 1

You have constructed, up to this point, a monomorphism $H_n C / \bigcap F_p H_n C \to \lim H_n(C/F_p C)$, and this is compatible with the map from $H_n C$. This allows you to construct the right-hand square, and then assemble all of, the following map of exact sequences: $$ \require{AMScd} \begin{CD} 0 @>>> \bigcap F_pH_n(C) @>>> H_n(C) @>>> H_n(C)/\bigcap F_pH_n(C) @>>> 0\\ @. @VVV @| @VVV @.\\ 0 @>>> \mathbin{\lim\limits_{\longleftarrow}}^1H_{n+1}(C/F_pC) @>>> H_n(C) @>>> \mathbin{\lim\limits_{\longleftarrow}} H_n(C/F_pC) @>>>0. \end{CD}$$

The middle map is an isomorphism and you have shown that right-hand vertical map is injective. Applying the snake lemma, we find that all the vertical maps are isomorphisms.