Elementary (?) question on differentials in a spectral sequence Suppose I have a chain complex $C$ with differential $D$ and filtration $F$.  Suppose further that I can decompose $D$ by its action on the filtration, i.e. there are maps $D = D_1 + D_2 + \cdots$ so that $D_k: F^iC \to F^{i+k}C$.  (In every example I am working with, the filtration is bounded.)
Now consider the spectral sequence $\{E^r, d^r\}$ induced by the filtration on $C$.  Is there an easy relationship between $d^r$ and $D_r$?
I know (say, from ncatlab) that $d^r$ is the restriction of $D$ to $r$-almost cycles, i.e. "elements in filtering degree $p$ on which $D$ decreases the filtering degree to $p-r$."  Can we say that $d^r$ is the map induced by $D_r$ on the appropriate quotient?

The answer is no!  One can see this by computing small examples in which $D = D_1 + D_2$ but the differential on $E^3$ is non-trivial.  It is true that $d^r$ is determined by $E_1, \ldots, E_r$ and $D_1, \ldots, D_r$.
