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I currently started studying spectral Sequences with the book of John Rognes. In his first chapter he defines Spectral Sequences as a sequence of bigraded abelian groups with differentials of certain degree with one condition. In the case that the spectral sequence stabilizes in each degree he defines the infinity page and defines what is means for a spectral sequence to converge to a graded filtered Group. Then he give a remark which should give me a motivation why spectral sequences are useful. I will just post it here:

Spectral Sequences - John Rognes

I dont get the end. At the end he just claims that we inductvly have to solve the extension problem

$$ 0\to F_{s-1}G_n \to ?? \to E^\infty_{s,n-s} \to 0$$

to get $F_sG_n$ and then inductivly recover $G_n$. But how do I do this? If I recall correctly my Algebra courses this is not even solvable in general as if $Ext^1(E^\infty_{s,n-s},F_{s-1}G_n) \not= 0$ there are at least 2 non isomorphic extensions, and I have no clue which one to choose? So am I confusing something or is something missing or what is the point in all of this?

user1072285
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    Well, sometimes only trivial extensions exist (when the Ext group you wrote vanishes, which is always the case over a field, for example). Even if you can’t solve the extension problem, the information a spectral sequence gives you (like the rank) is still extremely useful. – Frank Oct 11 '25 at 15:40

2 Answers2

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I assume you've already encountered things like the long exact sequence in homology of a short exact sequence $0\rightarrow C^{\prime}\rightarrow C\rightarrow C^{\prime\prime}\rightarrow 0$ of chain complexes. One might argue that this allows you to compute the homology of $C$ if you already know the homology of $C^{\prime}$ and $C^{\prime\prime}$, but then you may also object that this not true because there are extension problems. Nonetheless, I hope you don't need convincing that the long exact sequence in homology is something extremely useful. There are many reasons for this, of which I will list two:

  1. In many cases, the long exact sequence exhibits degenerate behavior. E.g. it tells you that $C^{\prime}$ is acyclic if and only if $C\rightarrow C^{\prime\prime}$ is a quasi-isomorphism. In this case, the extension problems trivialize.

  2. Even if you cannot solve the extension problems, the long exact sequence encodes them in a natural manner. That is, if the above short exact sequence of chain complexes maps to another short exact sequence of chain complexes $0\rightarrow D^{\prime}\rightarrow D\rightarrow D^{\prime\prime}\rightarrow 0$, then you can use the $5$-lemma to conclude that if two of the maps $C^{\prime}\rightarrow D^{\prime}$, $C\rightarrow D$ and $C^{\prime\prime}\rightarrow D^{\prime\prime}$ are quasi-isomorphisms, then so is the third. This is even more useful (and oftentimes underemphasized in introductions)!

Now, my final claim is that spectral sequences are the more powerful relatives of long exact sequences and any of the considerations expressed above apply as well to spectral sequences as they do to long exact sequences.

Thorgott
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    I understand your point. My problem is a bit then with SES there are easy examples where I immediatly see why the long exact sequence is usefull. I dont see "easy" examples of spectral sequences, and in fact in my book the first which will come up is the serre spectral sequence which is not easy. Is there any easy maybe nearly trivial exampel of a spectral sequence where I can see why extension problems are not always a problem? – user1072285 Oct 11 '25 at 21:04
  • @user1072285 for example calculating $\pi_4(S^3)$ with the Serre spectral sequence results in only one nonzero term on the relevant diagonal of the $E_{\infty}$ page (this is a homological spectral sequence). You can probably find proofs online, see https://math.stackexchange.com/questions/1102897/computing-pi-4s3-using-serre-spectral-sequence for example. – Steven Oct 11 '25 at 21:25
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    @user1072285 The 'easy' examples of stuff you can do with spectral sequences are a double-edged sword, because they are 'overkill'. Like, I can show you a neat spectral sequence argument for why cellular homology is the same as singular homology, but then you may rightfully object that you learned an elementary proof of that statement in an algebraic topology lecture that did not require all that complicated algebraic setup. The really fruitful applications of spectral sequences do require a bit of set-up. In topology, some computations with the Serre SS are go-to examples (see Hatcher's text) – Thorgott Oct 11 '25 at 21:39
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    In homological algebra, you may find it nice to apply a Künneth SS to prove that quasi-isomorphisms of chain complexes are stable under tensor products (surprisingly non-obvious in general, though you may object that you typically only consider chain complexes for which quasi-isomorphisms = homotopy equivalences). In algebraic geometry, you may like to use a Cech-to-derived spectral sequence to prove the Mayer-Vietoris sequence for sheaf cohomology or prove Leray's theorem (though you can do both by hand also). – Thorgott Oct 11 '25 at 21:44
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    In group cohomology, you can try using the Lyndon-Hochschild-Serre SS to get results on cohomological dimension, etc. Back to topology, I should at least also mention that you can get nice results out of the Atiyah-Hirzebruch spectral sequence (you can reprove the comparison theorem, though that can also be done by hand, but you can also compute extraordinary homology of, say, complex projective spaces). In general, I recommend you also check some other texts besides Rognes' (e.g. Hatcher's Spectral Sequences, McCleary's A User's Guide to Spectral Sequences) for more applications. – Thorgott Oct 11 '25 at 21:46
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Although, like Thorgott mentions, degeneracy often happens (especially in spectral sequence examples you see in courses) I think it is worth mentioning that extension problems can be "real problems":

It is often said that one of the big goals of homotopy theory is to compute the stable homotopy groups of spheres, $\pi_* S$. However, this is not what the professionals are trying to compute, nor is it what they want to compute: many would prefer to know the $\infty$-page of the Adams spectral sequence for the sphere instead. The difference is that this also records a filtration on $\pi_* S$; practically, this cuts up $\pi_* S$ into bite-sized pieces. Given the enormous complexity of $\pi_* S$, the helpfulness of cutting it up like this should not be ignored. The downside of the $\infty$-page is that it does not actually tell you $\pi_* S$: there are extension problems that one would have to solve. Tagging these on as additional information would be a holy grail for computational homotopy theory.

https://webspace.science.uu.nl/~0554804/seminars/syn-minicourse/lecture1.pdf

Steven
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