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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.

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    $\begingroup$ I don't know a good answer for this, but similar issues arise with respect to the Localized Parametrized Adams Spectral Sequence which arises in work on the Telescope Conjecture. I know that Ravenel spent a long time looking for a good abstract framework to encapsulate these issues, and never came up with something that he found satisfactory. So the problem may be hard, but a good answer might be important. $\endgroup$ – Neil Strickland Feb 24 '18 at 12:12
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    $\begingroup$ Maybe check out Deligne’s papers on mixed Hodges structures where he thinks about this, and Miller’s paper “on relations between Adams spectral sequences” where he gets at a differential this way. But yes, in general the relationship is not well understood computationally $\endgroup$ – Dylan Wilson Feb 24 '18 at 15:11
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    $\begingroup$ I dealt with a case of short exact sequences of spectral sequences in section 5 of folk.uio.no/rognes/papers/highfix.pdf (JPAA, 1999). A SES of E^r-terms leads to a LES of E^{r+1}-terms, but in my case this split into a SES of E^{r+1}-terms, and I could continue. $\endgroup$ – John Rognes Feb 25 '18 at 0:08

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