I am reading A User's Guide To Spectral Sequences and I don't understand the example in the informal introduction chapter:

We want to compute $H^*$, where $H^*$ is a graded $R$-module or a graded $k$-vector space [...] Suppose further that $H^*$ is filtered, that is, $H^*$ comes equipped with a sequence of sub objects $$ H^* \supset ... \supset F^n H^* \supset F^{n+1}H^* \supset ... \supset \{0\} $$ For the sake of clarity, let's assume for this chapter that $H^*$ is a graded vector space over a field $k$ and that $H^*=F^0 H^*$, that is, our filtration is bounded below by $H^*$ in the 0th filtration. For example [...] there is an obvious filtration, induced by the grading and given by $F^p H^*=\bigoplus_{n\geq p} H^n$.


A filtration of $H^*$, say $F^*$, can be collapsed into another graded vector space, called the associated graded vector space and defined by $E^p_0(H^*)=F^pH^*/F^{p+1}H^*$. In the case of a locally finite graded vector space (that is, $H^n$ if finite dimensional for each $n$), $H^*$ can be recovered up to isomorphism from the associated graded vector space by taking direct sums, that is $$ H\cong \bigoplus_{p=0}^\infty E^p_0(H^*) $$

I do not understand what is actually being computed here and from what. So, I see it, there is an unknown vector space $H^*$, what about $F^n H^*$ for $n>0$? Are these spaces given? Also, if we want to recover $H^*$ by taking direct sums as above, then don't we need to know what $H^*$ is, since the first summand is $\frac{F^0 H^*}{F^1 H^*}=\frac{H^*}{F^1 H^*}$, so we need to know that $H^*$ is?


In general, when using a spectral sequence, $H^*$ is the unknown piece. You then attempt to look at different parts of the spectral sequence. There will be groups $E_\infty^{pq}$, some of which may be easy to compute (the form of the spectral sequence gives you information on how to compute them). Each one is isomorphic to the quotient $F^pH^{p+q}/F^{p+1}H^{p+q}$. This is a piece of the graded vector space. You then look at different $E^{pq}_\infty$ groups to get different pieces of the graded vector space. For instance you could look at $E^{p+1,q-1}$. Each $E^{p+r,q-r}$ for $r\in\mathbb{Z}$ will give you a different piece of the same filtration quotient of $H^{p+q}$.

Hopefully, you can compute enough of the filtration via the $E_\infty$ groups to get some information about $H^*$, and sometimes you can completely recover $H^*$. This should become clear once you look at various examples.


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