# introductory book on spectral sequences

I have studied some basic homological algebra. But I can't send to get started on spectral sequences. I find Weibel and McCleary hard to understand.

Are there books or web resources that serve as good first introductions to spectral sequences? Thank you in advance!

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I like Wiebel immensely as well. But the most readable introduction I've seen to the topic is Bott and Tu's classic DIFFERENTIAL FORMS IN ALGEBRAIC TOPOLOGY. You can also try the nice presentation in the second edition of Joseph Rotman's homological algebra book.That should help you,Colin. –  Andrew L Apr 22 '10 at 17:40

Many of the references that people have mentioned are very nice, but the brutal truth is that you have to work very hard through some basic examples before it really makes sense.

Take a complex $K=K^\bullet$ with a two step filtration $F^1\subset F^0=K$, the spectral sequence contains no more information than is contained in the long exact sequence associated to $$0 \to F^1\to F^0\to (F^1/F^0)\to 0$$ Now consider a three step filtration $F^2\subset F^1\subset F^0=K$, write down all the short exact sequences you can and see what you get. The game is to somehow relate $H^*(K)$ to $H^*(F^i/F^{i+1})$. Suppose you know these are zero, is $H^*(K)=0$? Once you've mastered that then ...

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OK, I give up. The formulas look awful, but I'm not sure why. –  Donu Arapura Apr 22 '10 at 18:57
@Donu: I fixed it. As you can see, when the latex isn't displaying right for no apparent reason, you should try enclosing the expression in backticks, i.e., $\$ ... \'. –  Pete L. Clark Apr 22 '10 at 19:46
That last ' should rather be a . –  Pete L. Clark Apr 22 '10 at 19:47
OK, Pete. Thanks. –  Donu Arapura Apr 22 '10 at 20:39
Can't upvote this enough. The best way to come to grips with spectral sequences is to get your hands unrecognizably dirty with lots of examples and manual computation. –  Eric Peterson Apr 23 '10 at 1:20
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Bott and Tu, "Differential forms in Algebraic Topology" has some very nice exposition on spectral sequences. It has a fairly geometrical starting point, motivating the whole subject by generalizing the Meyer-Vietoris sequence to more complicated coverings and relating Cech cohomology to de Rham cohomology.

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I'm pleased to see Bott-Tu recommended; it's one of my favorite books. When I was a graduate student (that's thirty years ago!), the best, by far, exposition of spectral sequences was in Bott's course. This became part of the book by Bott and Tu, with, I believe, some help from Dan Freed who took Bott's course as an undergraduate. But since I'm far from an expert and have not read anything about spectral sequences since then, I was not sure whether something clearly better had appeared by now. –  Deane Yang Apr 22 '10 at 15:56
As you say, the motivation is important. I've recommended the book to people, only for them to jump to chapter 3, titled spectral sequences, but chapter 2 is already on the topic. –  Ben Wieland Apr 24 '10 at 22:33

I found Allen Hatcher's notes (which can be found here) clear and very helpful. They're not complete, but what is there is excellent, I think.

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You're right,but until they're finished,they're of limited help,Daniel. –  Andrew L Apr 22 '10 at 17:42
Andrew L, that's certainly false. –  Kevin H. Lin Apr 22 '10 at 18:01
I attended a seminar about spectral sequences which was partly based on Hatcher's notes. That worked pretty well! –  Ulrich Pennig Apr 24 '10 at 7:57

There's also a rather nice chapter in "The Heart of Cohomology" by Goro Kato. It could however be argued that I'm biased since I'm the one who typeset the book :)

Another good starting point is Timothy Chow's "http://www.ams.org/notices/200601/fea-chow.pdf".

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+1 Upvote for Chow's article. –  Colin Tan Apr 26 '10 at 13:22
+1 for "The Heart of Cohomology". It was even better in the Japanese original though... –  Daniel Moskovich Dec 4 '12 at 5:38

Also, Ravi Vakil has some nice notes about spectral sequences on his website: math.stanford.edu/~vakil/0708-216/216ss.pdf

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I would like to second this. It does not go to a lot of details, but the concrete explanation makes me less fearful of spectral sequence. –  Hailong Dao Apr 22 '10 at 18:43

Hi Colin,

I am appending a list of references I found useful.

MR0243527 (39 #4848) Mitchell, Barry . Spectral sequences for the layman. Amer. Math. Monthly 76 1969 599--605.

MR1721118 (2000m:55003) McCleary, John . A history of spectral sequences: origins to 1953. History of topology, QA611.A3 H57 1999 631--663, North-Holland, Amsterdam, 1999.

Matthew Greenberg:Spectral sequences http://www.math.mcgill.ca/goren/SeminarOnCohomology/specseq.pdf

Tom Weston:Inflation-Restriction sequence http://www.math.mcgill.ca/goren/SeminarOnCohomology/infres.pdf

Romero, Rubio, Sergeraert : Computing Spectral Sequences : http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Ana-JSC.pdf

David Brown Serre spectral sequences and applications http://www.math.wisc.edu/~brownda/math/papers/sss.pdf

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The book which cured my fear of spectral sequences is "Cohomology Operations and Applications in Homotopy Theory" by Moser and Tangora. It only touches applications in topology, and by todays standards it would be considered very basic; the upside of this is that a lot of the material is passed in the exercises (another upside is that it's \$10 on Amazon).

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I liked Mosher and Tangora quite a lot, too. An advantage of its concreteness and focus on specific applications is that there are lots of calculations which they do and which you, too, can do explicitly-- this is one of the subjects where it is hard to have a solid understanding without doing a fair number of examples and computations oneself. –  Allison Smith Apr 22 '10 at 16:36
I think the book is really pretty good at introducing spectral sequences because, like hatcher, it has an application/computation in mind that you can do right away. –  Sean Tilson May 20 '10 at 5:00

Tom Weston has a popular set of notes on spectral sequences on his website:

http://www.math.umass.edu/~weston/ep.html

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Seconded! I found these very helpful. –  babubba Apr 22 '10 at 17:32

I quite liked Gelfand-Manin's Methods of Homological Algebra. For a tranlsation and expansion of the section in EGA you would definitely like Daniel Murfet's notes.

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Chapters 9 and 10 of this shows how to apply spectral sequences to get many useful results rather than focusing on their construction and proofs of their properties.

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I couldn't agree with you more! Davis & Kirk are wonderful! –  Leon Lampret Jun 22 '13 at 10:40

I would recommend that everyone's very first (zeroth?) introduction would be Timothy Chow's excellent short article You Could Have Invented Spectral Sequences. It doesn't give a lot of technical details, but it will definitely remove your fear before you start on a more advanced exposition.

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There's always McCleary's "A User's Guide to Spectral Sequences".

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Apparently I should learn to read quesions... –  Simon Rose Apr 22 '10 at 17:36

Ken Brown's book, "Cohomology of groups" also gives a fairly readable introduction to spectral sequences. The algebra is kept fairly simple here, and most of the discussion is about computing the homology of a double complex, and constructing the Lyndon-Hochschild-Serre spectral sequence. So it doesn't go particularly deep, but it's a non-frightening place to start.

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I'm going to have to agree with everyone who recommends Bott & Tu. That provided me with a good understanding of the basic setup. After I was comfortable with that, I moved on to Hilton & Stammbach's book "A Course in Homological Algebra" that did a good job of showing how the general idea works for Abelian categories.

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