Suppose $(C,d)$ is a *chain complex*: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A *filtration* is an ascending chain of $d$-invariant subgroups

$0 = F_0 \subseteq F_1 \subseteq \cdots \subseteq C$.

It induces a filtration on the homology of $C$ (a complex with zero differential),

$0= H_0 \subseteq H_1 \subseteq \cdots \subseteq H(C,d)$.

In favorable situations, the associated graded group $\bigoplus (H_{i+1}/H_i)$ can be computed using a spectral sequence, and this gives us information about $H(C,d)$.

Suppose now we have another filtration of the *same* complex,

$0 = F_0' \subseteq F_1' \subseteq \cdots \subseteq C$.

*What assumptions are needed in order to incorporate this information into the spectral sequence, and how does one go about doing this?*

### A Motivating Example

Suppose you have a short exact sequence of double complexes, $0 \to (C'_{\bullet,\bullet},d') \xrightarrow{f} (C_{\bullet,\bullet},d) \to (C''_{\bullet,\bullet},d'') \to 0$

The middle complex $C$ is filtered as usual, by $F_i = C_{\leq i, \bullet}$ (I'm assuming here the differential $d = d_v + d_h$ decomposes as a sum of two differentials which commute up to some sign, where $d_v$ decreases the first grading by one and preserves the second, and $d_h$ does the opposite).

By assumption, the map $f$ commutes with $d_v,d_h$ and preserves the grading, so we can consider $0 \subseteq C' \subseteq C$ as a second filtration on $C$. To see that this has interesting information, note that the first page of the spectral sequence associated with $F$ has $H(C_{\bullet,\bullet},d_h)$ with differential $d_v$, and similarly for the first pages of the spectral sequences for $C'$ and $C''$ with their grading-filtrations. These first pages sit in a long exact sequence, which may be computationally useful in itself - but it seems there's more information to be exploited here.