Let $(K,D)$ be a differential complex of abelian groups, and $K = K_0 \supset K_1 \supset K_2 \supset \cdots \supset K_{p+1} = 0$ a filtration of $K$ by sub-complexes. Let $(E^{r},d^r)_{r\ge 1}$ be the spectral sequence associated to this filtration. Let $1 \le r \le p$ an integer and suppose $\xi \in E^r$ such that I can find $x \in K_{p-k}$ for some $0 \le k \le p$ such that $D(x) = 0$ and $x$ is a representative for $\xi$ in the succesives quotients.
Is it true that $d^r(\xi) = 0$ ?
I found this statement in http://arxiv.org/abs/1411.1685 in page 8. In the notations I use here the statement goes like : Indeed, closed (in $(K,D)$) representatives for elements in $E^2$ may be given, and hence all higher differentials are $0$.
What theorem are they using here?
