Infinite dimensional homology and spectral sequences

Question

I am new to spectral sequences, so I'm not sure about the difficulty of this question.

Suppose we have a filtration of a chain complex $\emptyset=D_{-1}\subset D_{0}\subset D_1\subset\dots D_n= C$ and we want to study the homology groups $H_n(C)$. Of course the most natural way is to study the spectral sequence associated to the filtration. In the case I'm working I am able to undestand $D_i/D_{i-1}$ and $H_n(D_i/D_{i-1})$ which give us the first two sheets of the spectral sequence, in particular I know when those homology groups have infinite dimension.

My question is whether knowing that $\dim E_{p,q}^r=\infty$ for some $(p,q)$ allow us to say that $\dim H_n(C)=\infty$. Maybe this is true by adding extra hypothesis, If so I would like to know it too. My only guess is that, maybe, by induction we could prove that $\dim E_{p,q}^r=\infty$ implies $\dim E_{p',q'}^{r+1}=\infty$ for some $(p',q')$, which would imply that $\dim E_{p'',q''}^\infty=\infty$ for some $(p'',q'')$ and since $E^\infty_{p'',q''}$ is a section of $H_n(C)$ this would prove our claim, but I'm not sure whether this induction true or not.

Answer 1

Your problem is that the differentials in the spectral sequence can "wipe out" infinitely much stuff. If you know that $\dim E^r_{p,q}=\infty$ and that $E^r_{p',q'}$ is finite dimensional for all $(p',q')$ that are possible sources or sinks of the differentials $d^s$ for $s\geqslant r$ ending or beginning respectively at $(p,q)$, then you're in good shape.