Spectral sequences and filtrations

Definition: Let $C^\bullet\in C(A)$ be a cochain complex with boundary maps $d^\bullet$ over some category $A$. A filtration of $C^\bullet$ is a sequence of objects $F^nC^\bullet$ with boundary maps $d^{\bullet,n}_h$ in the category of cochain complexes $C(A)$ of $A$, either a
$$\begin{align*} \text{decreasing filtration}\ C^\bullet & \supseteq \cdots \supseteq F^{n-1}C^\bullet \supseteq F^nC^\bullet\supseteq F^{n+1}C^\bullet\supseteq \cdots\ \text{or}\ \\ \text{increasing filtration}\ C^\bullet & \supseteq \cdots \supseteq F^{n+1}C^\bullet \supseteq F^nC^\bullet\supseteq F^{n-1}C^\bullet\supseteq \cdots, \end{align*}$$
where ``$\supseteq$" is defined as necessary, along with maps $d_v^{\bullet,n}:F^nC^\bullet \to F^{n\pm1}C^\bullet$. These maps are compatible, in the sense that $d_v^{k\pm1,n}d_h^{k,n} = d_h^{k,n\mp1}d_v^{k,n}$.

Example: Define ``$\supseteq$" as $X\supseteq Y$ iff $\Hom(Y,X)$ is non-empty. The bete (or brutal) filtration of $C^\bullet$ is a decreasing filtration
$$\left(F^nC^\bullet\right)^i = \begin{cases} 0 &\ \text{if}\ i<n, \\ C^i & \ \text{if}\ i\geqslant n,\end{cases} \hspace{1cm}\text{with}\hspace{1cm} H^k(F^nC^\bullet) = \begin{cases} 0 & \ \text{if}\ k<n \\ Z^n & \ \text{if}\ k=n, \\ H^k(C^\bullet) & \ \text{if}\ k>n.\end{cases}$$
This filtration may be represented by the diagram

which clearly commutes. The good filtration of $C^\bullet$ is also a decreasing filtration
$$\left(F^nC^\bullet\right)^i = \begin{cases} C^i &\ \text{if}\ i<n, \\ Z^iC^\bullet & \ \text{if}\ i=n, \\ 0 & \ \text{if}\  i>n,\end{cases} \hspace{1cm}\text{with}\hspace{1cm} H^k(F^nC^\bullet) = \begin{cases} H^k(C^\bullet) & \ \text{if}\ k\leqslant n, \\ 0 & \ \text{if}\ k>n.\end{cases}$$
This filtration may be represented by the diagram

which also commutes. Both of these are also called truncations. The good filtration is "better" because the cocycle groups $Z^n$ do not appear in the cohomology groups. The same may be done for homology groups.

Definition: Set $F^nC^k = (F^nC^\bullet)^k = F^nC^\bullet \cap C^k$, and let the zeroth page of the cohomology spectral sequence of $C^\bullet$ with the filtration $F$ be given by
$$\begin{align*} E^{p,q}_0 & = F^pC^{p+q} / F^{p+1}C^{p+q} \ \ \text{if $F$ is decreasing,} \\ & = F^pC^{p+q} / F^{p-1}C^{p+q} \ \ \text{if $F$ is increasing.} \end{align*}$$
Let the first page of the cohomology spectral sequence of $C^\bullet$ with the filtration $F$ be given by
$$\begin{align*} E^{p,q}_1 & = H^{p+q}(F^pC^\bullet / F^{p+1}C^\bullet) \ \ \text{if $F$ is decreasing,} \\ & = H^{p+q}(F^pC^\bullet / F^{p-1}C^\bullet) \ \ \text{if $F$ is increasing.} \end{align*}$$
From now on, assume that $F$ is an increasing filtration. Let the second page of the cohomology spectral sequence of $C^\bullet$ with the filtration $F$ be given by
$$E^{p,q}_2 = \frac{\ker(E_1^{p,q} \to E_1^{p+1,q})}{\text{im}(E_1^{p-1,q}\to E_1^{p,q})}.$$
Continue in this manner and let the $r$th page of the cohomology spectral sequence of $C^\bullet$ with the filtration $F$ be given by
$$E_r^{p,q} = \frac{\{x\in F^pC^{p+q}\ :\ dx\in F^{p+r}C^{p+q+1}\}}{F^{p+1}C^{p+q} + dF^{p-r+1}C^{p+q-1}}.$$

The same may be done for a homology spectral sequence. Note that a spectral sequence may also be defined without coming from a filtration.

Definition: A homology spectral sequence is a collection of objects $E^r_{p,q}$ and maps $d^r_{p,q}:E^r_{p,q}\to E^r_{p-r,q+r-1}$ with $d^rd^r=0$ such that
$$E^{r+1}_{p,q}\cong \ker(d^r_{p,q})/\text{im}(d^r_{p+r,q-r+1}).$$
Similarly, a cohomology spectral sequence is a collection of objects $E_r^{p,q}$ and maps $d_r^{p,q}:E_r^{p,q}\to E_r^{p+r,q-r+1}$ with $d^rd^r=0$ such that
$$E_{r+1}^{p,q}\cong \ker(d_r^{p,q})/\text{im}(d_r^{p-r,q+r-1}).$$

References: Weibel (An introduction to homological algebra, Chapter 1.2), McCleary (A user's guide to spectral sequences, Chapter 2.2), Hutchings (Algebraic topology lecture notes, see math.berkeley.edu/~hutching/teach/215b-2011)