Suppose $\{(E\_r,d\_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E\_r,d\_r)$ is a chain complex over $\mathbb{F}$ and $E\_{r+1}=H(E\_r,d\_r)$. For simplicity, suppose $E\_N=E\_{N+1}=\cdots=E\_{\infty}$ for some fixed number $N$. I have the following questions:


1. Can we construct a filtered chain complex $(C,d)$ such that the induced spectral sequence is $\{(E\_r,d\_r)\}$?
2. Can we determine $(C,d)$ up to some equivalence (like filtered quasi-isomorphism)?
3. Can we relax the assumption to obtain the similar results?
4. Are there any good references?


Indeed, the spectral sequence in my mind is from the unrolled spectral sequence in


<https://ncatlab.org/nlab/show/conditional+convergence>


There are many similar questions, which gives some potential counterexamples when we replace $\mathbb{F}$ by $\mathbb{Z}$ or without the assumption about the convergence at some finite page $E\_N$.


[Complete invariant of filtered chain complexes under chain homotopy equivalence](https://mathoverflow.net/questions/350434/complete-invariant-of-filtered-chain-complexes-under-chain-homotopy-equivalence)


[Functoriality of filtered spectral sequences](https://mathoverflow.net/questions/345676/functoriality-of-filtered-spectral-sequences)


[isomorphic spectral sequences => quasi-isomorphic filtered chain complexes?](https://mathoverflow.net/questions/100157/isomorphic-spectral-sequences-quasi-isomorphic-filtered-chain-complexes)



Can we construct a filtered chain complex from a spectral sequence?
1. Yes, since you're working over a field, you can decompose your spectral sequence into a big direct sum of


a) permanent cycles, and


b) for each r>1, chains supporting a d\_r, and the cycle hit by that d\_r.


It is straightforward to represent each such summand by a simple filtered chain complex, so do so, and then take a direct sum. The resulting filtered chain complex has its spectral sequence isomorphic to the one you began with. I don't know a written reference for this, but it is an old idea, and it is not difficult to write out the details.


2. I think this question is addressed by the more general results in "Model category structures and spectral sequences" by Cirici, Santander, Livernet, and Whitehouse: <https://arxiv.org/pdf/1805.00374.pdf>
3. If I recall correctly, you can replace your ground field with any commutative ring without changing the answer to #1. But I am only going from memory here!
4. See answers to #1 and #2.


You mentioned unrolled exact couples. Let me mention that question #2 becomes much, much more subtle if you work with exact couples rather than filtered chain complexes, but at least question #1 still has the same answer (for basically the same reasons) in that setting.


