Spectral sequence of a filtered complex: convergence conditions and abelian categories

Question

There is a theorem that if given a filtered complex and the filtration is bounded then there is a spectral sequence whose 0th and 1st page have specific forms and the sequence converges to (co)homology of the complex. I followed the proof in Serge Lang's book, Algebra and I asked myself two questions yet unanswered:

1. Does the filtration have to be bounded for the spectral seq. to converge?

2. The proof seems to take place in category of modules, not in a general abelian category. I tried to generalize the statement somehow using Freyd Mitchell embedding thm, but it wasn't easy. Does one have to be able to do 'diagram-chasing' in order to prove the theorem?

Thank you.

Answer 1

For the convergence issue, a possible reference is the chapter 3 of McCleary's A User's Guide to Spectral Sequences. For example you have the following theorem (I will explain the terms), for a cocomplex and an increasing filtration:

Theorem. Let $(A,d,F)$ be a filtered dg-module such that the filtration is exhaustive and weakly convergent. Then the associated spectral sequence with $E^{p,q}_1 = F^p H^{p+q}(A,d) / F^{p+1} H^{p+q}(A,d)$ converges to $H(A,d)$.

Here a filtration is said to be:

• exhaustive if $A = \bigcup_s F^s A$;
• weakly convergent if for all $p$, $$Z^p_\infty = \bigcap_r Z^p_r \iff F^p A \cap \operatorname{ker} d = \bigcap_r (F^p A \cap d^{-1} (F^{p+r} A)).$$

This is one possible example of a convergence theorem. In general you do need conditions on the filtration to ensure convergence; boundedness is just one that is simple to state. You can easily construct examples of spectral sequences that don't converge / converge to nonsense if you remove these conditions, for example if I set $F^p A = 0$ for all $p$ I will obviously not get anything interesting; if I set $F^p A = A$ for all $p \in \mathbb{Z}$ another kind of pathology occurs.

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It is possible to work with spectral sequences in general abelian categories. One of the most important application is the Grothendieck spectral sequence that generalizes many other examples (the Leray SS, the universal coefficient SSs, the Serre SS...). There are a few references given in McCleary's book in chapter 12 under the heading "Abelian categories". In particular he cites Gelfand and Manin's Methods of homological algebras; but I recommend that you familiarize yourself thoroughly with spectral sequences in modules before attacking general spectral sequences.