# 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!

• 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. Apr 22, 2010 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 ...

• 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. Apr 23, 2010 at 1:20
• @EricPeterson: The problem is that there aren't really that many examples around. For the Leray-Serre spectral sequence, you need a fibration, and there aren't that many of them around, in which you can compute the (co)homology of two out of three spaces.
– Leo
Jun 22, 2013 at 9:36
• Indeed. I started to learn with a three-step-filtration-like SS that showed up in my research during my PhD student times. Dec 7, 2021 at 9:10

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.

• 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. Apr 22, 2010 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. Apr 24, 2010 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.

• You're right,but until they're finished,they're of limited help,Daniel. Apr 22, 2010 at 17:42
• Andrew L, that's certainly false. Apr 22, 2010 at 18:01
• I attended a seminar about spectral sequences which was partly based on Hatcher's notes. That worked pretty well! Apr 24, 2010 at 7:57

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

[EDIT by DG: These notes have meanwhile become (in an updated version) a section (currently 1.7, as of 29 December 2015) of Ravi Vakil's The Rising Sea: Foundations Of Algebraic Geometry Notes.]

• 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. Apr 22, 2010 at 18:43
• These notes are particularly good for somebody like me, who hasn't (yet) needed to use the full power of spectral sequences, but just use them to do unpleasant homological algebra involving far too many cohomology groups in an efficient way. Vakil's spectral sequence proofs of the snake lemma (despite the typo, fixed in the version of this article that appears as 1.7 in Foundations of Algebraic Geometry) and five lemma were a massive eureka moment for me, in terms of understanding the general principle. (They also helped me actually remember the methodology, rather than look it up every time.) Apr 28, 2016 at 10:12

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".

• +1 Upvote for Chow's article.
– user2529
Apr 26, 2010 at 13:22
• +1 for "The Heart of Cohomology". It was even better in the Japanese original though... Dec 4, 2012 at 5:38

The book which cured my fear of spectral sequences is "Cohomology Operations and Applications in Homotopy Theory" by Mosher 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).

• 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. Apr 22, 2010 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. May 20, 2010 at 5:00

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 https://www.math.emory.edu/~dzb/math/papers/sss/

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

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

• Seconded! I found these very helpful. Apr 22, 2010 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.

If you like David Mumford's writing style, i.e. clear, succinct and authoritative, try his notes in AG2:

http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap7-8.pdf

Chapters 9 and 10 of James F. Davis, Paul Kirk, Lecture Notes in Algebraic Topology, GSM #35, AMS shows how to apply spectral sequences to get many useful results rather than focusing on their construction and proofs of their properties.

• I couldn't agree with you more! Davis & Kirk are wonderful!
– Leo
Jun 22, 2013 at 10:40

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.

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

• Apparently I should learn to read quesions... Apr 22, 2010 at 17:36

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.

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.

My Knowledge of spectral sequences is coming from the chapter 2 of book "Algebraic geometry" by Fu Lei. I think it is detailed and very easy to read. It is not theoretical oriented by organized for application of spectral sequences.

The more nice I remember was also my first meet by HA: Rotman "Introduction to homological ALgebra"

This is probably irresponsible but I decided to place a link to a raw pdf file entitled "Self-dual formulation of homology, exact couples and spectral sequences". It is a beamer presentation and it is awful: it does not contain a single word, just a pretentious panoply of diagrams.

The reason I decided to do this: I think the primary drawback of spectral sequences is the nightmare of indices, which is moreover not justified at all. The essential content of spectral sequences does not have anything to do with gradings a priori. So I decided to work out the construction of exact couples in the non-graded setup.

The result was spectacular in one respect: it shows that even if you get rid of gradings, the thing is still quite complicated :D

Still, I find it amusing to see what remains when you throw all the grading stuff out.

• These diagrams are absolutely gorgeous. Mar 4, 2021 at 16:13
• @DanRamras Thank you! I wish I made something useful with them... Mar 4, 2021 at 17:37
• Bob Bruner had some spectral sequence diagrams (of the graded variety) displayed in an art installation somewhere. I'm not sure if that counts as useful :) Mar 4, 2021 at 17:58

There are Uwe Jannsen's lecture notes on étale cohomology: http://www.mathematik.uni-regensburg.de/Jannsen/Etale-gesamt-eng.pdf Spectral sequences are covered in section 6.