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When you first learn about rings, it's important to have examples of, say, a PID which is not a Euclidean domain, a UFD which is not a PID, and so forth, to help build intuition and provide test cases.

Well, I'm just starting to learn about spectral sequences. According to Eisenbud's treatment of spectral sequences, most everyday spectral sequences come from exact couples, most exact couples come from monic endomorphisms of differential modules, and I guess the next steps are differential modules with filtrations and gradings.

Is there a nice list somewhere with examples of, say, a spectral sequence that doesn't come from an exact couple, and so forth?

Of course, feel free to simply provide such examples here as well. :)

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From Wikipedia: "In fact, all known spectral sequences can be constructed using exact couples." (In my opinion, the word "known" in this sentence is a bit odd. Are there unknown spectral sequences that do not come from exact couples? If one does exist, it seems like it would be something contrived and not very helpful towards understanding spectral sequences.)

However, there are some spectral sequences that don't require knowledge of exact couples (and probably better understood w/o exact couples). One is the Grothendieck spectral sequence. This gives you information on how two (right) derived functors compose. The setup is this: Suppose $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are abelian categories and $\mathcal{A}$, $\mathcal{B}$ with enough injectives (think $\bf{R-mod}$ or $\text{Sheaves}(X)$), $F:\mathcal{B} \rightarrow \mathcal{C}$ and $G:\mathcal{A} \rightarrow \mathcal{B}$ left exact functors. One last thing: assume $G$ takes injective objects to $F$-acyclic objects. Then there is a convergent first quadrant spectral sequence for each object $A$ of $\mathcal{A}$:

$$E^{pq}_2 = (R^pF)(R^qG)(A) \Rightarrow (R^{p+q}FG)(A) $$

(There is a dual statement for right exact functors.)

You can use this to study base change with respect to Ext and Tor. The Leary spectral sequence is another example (this looks at what happens to global sections after applying the direct image functor). Another special case is the Lyndon-Hochschild-Serre spectral sequence. This appears in group cohomology. It allows you to compute the cohomology of $G$ (with coefficients in $A$) knowing the cohomology of $G/H$ with coefficients in $H^*(H,A)$. This provides a neat way of computing (for example) the cohomology of the dihedral groups or of the discrete Heisenburg group.

I'm not sure if this is exactly what you are looking for, but I think it is at least relevant. My understanding is that spectral sequences in algebraic topology usually involve exact couples, but spectral sequences in commutative algebra usually do not.

All of the information above is treated in Weibel's An Introduction to Homological Algebra. I really like this book. In his spectral sequence chapter, he develops the theory without use of exact couples. He doesn't even talk about them until the very end of the chapter.

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  • $\begingroup$ I imagine the bit about 'all known spectral sequences come from exact couples' just means 'all spectral sequences used in practice.' $\endgroup$ – Daniel McLaury May 29 '13 at 2:50

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