Functoriality of filtered spectral sequences What is the appropriate functoriality statement of a filtered chain map between filtered spectral sequences?
Suppose that we have two filtered chain complexes $C,C'$ and a filtered chain map $f\colon C\to C′$. This induces a map between the spectral sequences of the filtered chain complexes, and my question is what map does this induce on each page.
In particular, what relation, if any, is there between these induced maps, and the map that $f$ induces on the total homology of the complexes $f_\ast\colon H(C)\to H(C′)$? Is there a way that the associated graded of $f$ is related to the above? Is there any prefered reference for these matters?
 A: Let us use increasing filtrations $F^pC \subset F^{p+1}C\subset \ldots$. Then the homology of $C$ and $C'$ also inherit the structure of filtered modules by $F^pH(C) = im(H(F^p(C)\rightarrow C))$, or in words: a homology class in $H(C)$ has filtration degree $p$, if there is a representing chain in $F^pC$. With this structure $H(f):H(C)\rightarrow H(C')$ is filtration preserving; and the subquotients $F^{p}H_q(C)/F^{p-1}H_q(C)$ are exactly the entries of the $E_\infty$-page.
The groups arising in the finite pages are the groups of $r$-almost cycles divided by the group of $r$-almost boundaries (that name is really misleading; they are really honest boundaries, but there are more boundaries then just them). To simplify, let us use the notation $Z^{p,r}_q$ for all the chains whose boundary lives $r$ steps further down in the filtration, i.e. 
$$Z^{p,r}_q=\{x\in F^{p}C_q\mid dx \in F^{p-r}C_{q-1}\}.$$
Then the groups appearing on the $r$-th page of the spectral sequence are 
$$\frac{Z^{p,r}_q}{Z^{p-1,r-1}_q+d(Z^{p+r-1,r-1}_{q+1})}.$$
Informally an element is represented by a chain whose boundary lives $r$ steps further down in the filtration and we divide out things one step further down in the filtration and boundaries of chains living at most $r-1$ steps further up in the filtration.
Now the filtration preserving map induces maps $Z^{p,r}_q\rightarrow Z'^{p,r}_q$ and these induce map on the quotients above. This is the map that $f$ induces on finite pages. It sometimes differs from author to author (and application to application) which index convention is used. So the groups given above really arise somewhere, but where exactly depends on conventions. 
EDIT: I said I used increasing filtrations, but the signs in my definitions were for decreasing filtrations. I changed that.
