It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. <https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples>). My question is:under which conditions does this spectral sequence degenerate at $E\_1$ (actually, I am more interested in necessary conditions)? This condition appears to be equivalent to the image of the morphism $f\_1:E\_1\to D\_1$ lying inside all levels of the filtration of $D\_1$ by $g\_1^i(D\_1)$ (here I ignore the upper indices, and $g\_1^i$ denotes the $i$th iterate of $g\_1:D\_1\to D\_1$). Is this true; are there any references for this fact? 


Actually, I would like to conclude that $f\_1=0$; does this follow from the degeneration at $E\_1$? I am more interested in the bounded case; so an answer to the first part of my question is sufficient for my purposes; yet are there any other conditions ensuring that $f\_1=0$ whenever the spectral sequence degenerates at $E\_1$?



For which exact couples do associated spectral sequences degenerate at $E_1$?
Degenerating at $E\_1$ is, as you describe, equivalent to the image of the map $f\_1: D\_1 \to E\_1$ being contained in the image of $g\_1^i$ for all $i$. Roughly, this is because the definition of the $d\_r$-differential on a class $x$ is the equivalence class of any element $h\_1 g\_1^{1-r} f\_1(x)$, where $h\_1$ is the map $D\_1 \to E\_1$. If the spectral sequence degenerates, then $E\_1 = E\_r$ for all $r$ and so we don't need to worry about equivalence classes: then this asks that $g\_1^{1-r} f\_1(x)$ is in the kernel of $h\_1$, hence the image of $g\_1$. Then we repeat.


I do not know a place in the literature where this is explicitly stated.


It is not necessarily the case that $f\_1 = 0$ if the spectral sequence degenerates. For example, if the map $h\_1$ is zero so that $0 \to E\_1 \xrightarrow{f\_1} D\_1 \to D\_1 \to 0$ is an exact sequence, the spectral sequence still degenerates at the $E\_1$-page. This is not uncommon behavior for degenerating spectral sequences in cohomology, rather than homology.


