# Reference for spectral sequences

What are good expositions of spectral sequences, which include a thorough introduction to the topic as well as the most important examples of applications - maybe with an emphasis an topological and/or operator algebraic uses.

I know that the wikipedia article is quite extensive, but I'd like to have something... nicer.

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I think that spectral sequences are one of those things which it is good to spend some time bashing out oneself. In other words, rather than necessarily finding the best possible exposition, just find some exposition (surely Weibel has one) --- the key point will be the spectral sequences arising from a double complex. One can then take some paper which makes a spectral sequence argument, and try to figure out how Weibel (or whoever's) general nonsense meshes with the example at hand. Doing this a couple of times will turn you from a novice to an expert! –  Matt E Oct 27 '10 at 4:52
P.S. Sorry if the above is a little dogmatic. I agree that if you want to learn particular spectral sequences (especially not so obvious ones) then a nice reference is helpful. In my defense, this was a topic I struggled with for a long time in grad school, and finally mastered (to whatever extent I have mastered it) by following some version of the plan outlined above. –  Matt E Oct 27 '10 at 4:53
@Qiaochu: Thanks, I should have looked there before asking. –  Rasmus Oct 27 '10 at 16:06
I agree with Matt E on this. When I set out to understand spectral sequences I found all the available expositions full of conventions that I found distasteful. So I set myself the task of trying to compute the homology of a filtered complex in the most naive way possible, and see where it took me. Needless to say, I'm comfortable with the spectral sequence of a filtered complex now and I found this approach quite satisfying. –  Ryan Budney Nov 9 '10 at 6:39

I really like the way Bott & Tu develop spectral sequences in "Differential Forms in Algebraic Topology". It would be an especially good choice if you happen to have some familiarity with de Rham cohomology; they begin by just sort of easing you into the whole idea by proving the isomorphism of de Rham cohomology and Cech cohomology by constructing a double complex whose cohomology is isomorphic to both of those.

Of course, they only give uses for spectral sequences in topology. I think they mainly introduce them to be able to give the spectral sequences for the homology and cohomology of the total space of a fibration. In this application there is particularly nice geometric intuition for the algebraic machinery, which I found very useful for trying to get to the bottom of what these things really do.

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The exposition of spectral sequences in Switzer's "Algebraic Topology" is surprisingly readable, too. I just read it, and I like it a lot. –  Aaron Mazel-Gee Nov 14 '10 at 19:31

I like the exposition in the appendix to Eisenbud's Commutative Algebra, which does everything in a kind of friendly way via exact couples. It doesn't do the Grothendieck spectral sequence, though, but that's in many other places (e.g. Lang's Algebra), and essentially follows (after some homological stuff) from the spectral sequence of a double complex. (I should confess that I've not myself needed to know too much about spectral sequences at the moment (beyond the Leray spectral sequence and the Cech-to-derived functor one in sheaf cohomology), and my knowledge of them is correspondingly limited.)

EGA III has several applications of spectral sequences in algebraic geometry. The very first result, that the higher quasi-coherent cohomology vanishes on an affine scheme, implicitly uses the Cech-to-derived functor spectral sequence since the computation given is that of the Cech cohomology (but he doesn't actually go through the details--I think they're in Godement's Theorie de faisceaux, but the deduction of it as a special case of the Grothendieck's spectral sequence is in Milne's online notes on etale cohomology). Another application in EGA III is the proof of the proper mapping theorem: that a proper map of noetherian schemes preserves coherence. The argument is very pretty, and starts in the standard case by doing this for a projective morphism (which is essentially the calculation, as in Hartshorne, of the cohomology of the standard line bundles on $\mathbb{P}^n$). But to deduce it for a proper morphism, the real fun starts when you "approximate" a proper $f: X \to Y$ morphism by a projective one (using Chow's lemma) and then show that there is a "significant" sheaf on $X$ which pushes forward to $Y$ coherently using the projective morphism and the Grothendieck spectral sequence between various push-forwards. (There's also a devissage argument that I'm skipping here.)

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If you know a bit of algebraic topology Hatcher's book mentioned above is great. I thought it was a little slow going, so instead I prefer Mosher and Tangora, which is now a Dover book!

I think there are some great exercises that really made me feel comfortable with Spectral sequences. The first Two SS I would look at are Bockstein and Serre. First look at the description/construction/definition of $sq^1$ in MT. It IS the first differential in the mod two bockstein SS. so now you can compute integral cohomology from mod two cohomology once you know how $sq^1$ acts (this also shed light on $sq^1$ for me). To do these computations you should adjoin some indeterminate and use adams grading. then $d_1$ goes up one and to the left one.

Now you can use the serre SS and various simple fibre sequences to compute the cohomology of all the complex and real projective spaces WITH their ring structure! (first compute it for the infinite cases then restrict). This is all spelled out in MT (maybe not the part about restricting to get the finite dimensional projective spaces).

Then compute the cohomology of $\Omega S^n$ for every n, with ring structure!

Then prove the Thom Isomorphism Thm with a relative form of the Serre SS.

Also, when you are looking at the bockstein SS draw one of boardman's unrolled exact couples and look at what is going on. And dont use their construction of the steenrod operations, there are better places imo.

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Thank you for this very concrete advice! –  Rasmus Dec 2 '10 at 7:21
when you want to understand where the differential is coming from check out Conditionally convergent SS by Boardman. It is not a good idea to do this before the above though. –  Sean Tilson Dec 2 '10 at 20:10

I really like the last chapter of Aluffi's book "Algebra: Chapter 0."

Perhaps one of the canonical references (especially to see tons of applications) is John McCleary's "A User's Guide to Spectral Sequences."

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A nice place to start is Timothy Chow's "You Could Have Invented Spectral Sequences".

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Allen Hatcher has some textbooks available for free download on his website, including the first 100 pages of an unfinished text on Spectral Sequences in Algebraic Topology.

It should qualify as a thorough first introduction to the topic, but I don't think all of the important examples have made it into the text. I haven't read it myself, but I remember his more basic text on algebraic topology being very friendly and a good introduction.

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I really like the book J.Davis and P.Kirk, "Lecture notes in Algebraic topology".

In their book (in Chapter 9,10), they give a friendly and thorough introduction to Spectral Sequence with

1. full-detailed calculations
2. elegant interpretations
3. vivid examples.

Here are some nice interpretations and jokes which I like it.

Computing with spectral sequences is somewhat like computing integrals in calculus - in page 237 (introduction part of spectral sequence).

As a service to the reader, we will explicitly unravel the statement of the above theorem... - in page 242, giving detailed and helpful explanation.

We will use the following lemma, which shows why Serre classes are tailor-made to be used with spectral sequences. - in page 271.

Unfortuantely, there are so many errata in their book. The collection of errata is available from J. Davis's page. Actually, I read it from cover to cover. But, I remember that there are some errata not in J.Davis's collection.

Anyway, I would like to emphasize

J.Davis and P.Kirk book is a perfect introduction to Spectral Sequence.

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I strongly recommend reading Ravi Vakil's SPECTRAL SEQUENCES: FRIEND OR FOE? as a first introduction to spectral sequences. I have tried to learn spectral sequences elsewhere, but this had helped me to get a much better grasp at what is going on there..

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