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.



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


