tips on cohomology for number theory

I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings.

Do people just remember all the rules and go through the formal manipulations of the cohomology groups of class field theory mechanically, or are people actually "feeling" what is going on here. If it is the latter, could you give a list of mnemonics, or cheat-sheet, or a little fairy tale involving all the characters, so that it is easy to associate with the abstraction. Is there some strong intuition coming from algebraic-topology that would help here? Do people think of Cech covers when they do diagram chasing on crazy grids of exact sequences?

I personally was extremely happy with the central simple algebra approach to class field theory, because it gave me a whole new kind of creature, a CSA, which I could learn to adapt to and love. On the other hand, I don't see myself ever making friends with a 2-cocycle.

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Probably a candidate for community wiki. My 2 cents: I was very happy to learn about derived functors, they made the cohomological stuff "feel right". –  Konrad Voelkel Dec 11 '09 at 21:28
I think that the use of profinite group cohomology in class field theory is not the best place to start learning cohomology, as it is very specialized and complicated. I would start with algebraic and differential topology and then go to abstract algebra: Lie algebra cohomology, group cohomology, derived functors, spectral sequences, Ext and Tor, etc. This would allow you to "adapt to and love" things cohomological, and know them when you see them. –  Leonid Positselski Dec 11 '09 at 21:50
"fairy tale","extremely happy with the central simple algebra approach","adapt and love","making friends with a 2-cocycle". It is really wonderful to see you relate to mathematics with such passion: +1 and all my best wishes to this enthusiastic fellow. –  Georges Elencwajg Dec 11 '09 at 22:01
At some point you can make friends with etale cohomology, which is the basic cohomology theory in arithmetic geometry. Etale cohomology provides a bridge between Galois cohomology in number theory and singular cohomology in differential geometry. This means that on one hand etale cohomology over complex numbers coincides with singular cohomology. On the other hand Galois cohomology of Gal(F) is etale cohomology of a "single point" Spec(F). –  Evgeny Shinder Dec 11 '09 at 23:17
I just make it up as I go along everytime... –  some guy on the street Dec 12 '09 at 0:00
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5 Answers

Many number theorists, including me, learned Galois cohomology first via the proof of the Mordell-Weil theorem. The last chapter of Joe Silverman's book The Arithmetic of Elliptic Curves is a good source for this. It's very concrete and when you understand the proof you'll understand a lot about why number theorists like the cohomological formalism.

Edit: But note that you'll learn just about H^1, not anything higher. It's a start!

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Two-cocycles turned up, and were used in the study of central simple algebras before group cohomology was defined. The use of group cohomology greatly simplifies the classification of of central simple algebras over $\mathbb{Q}$, for example. Moreover, once you restate the classification (the Albert-Brauer-Hasse-Noether theorem) more abstractly in terms of group cohomology, you can apply it elsewhere.

Homology/cohomology is just like anything else in mathematics. At first it may seem strange, but once you understand it, it becomes familiar.

In algebraic number theory and class field theory, you mainly need the cohomology of finite groups, which in fact is a very good place to start, so my advice would be to first study thoroughly the cohomology of finite groups, as for example, in Part 3 of Serre's book Local Fields (Corps Locaux). From there, it is not difficult to understand the cohomology of profinite groups.

Loosely speaking, etale cohomology unites the usual cohomology of manifolds with the cohomology of the Galois groups of fields, and is of fundamental importance in arithmetic geometry.

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I learnt cohomology this way---first some of the topological/diff manifold theories (de Rham, singular), and then group cohomology. My memory at the time was that I found group cohomology incredibly weird after all the singular cohomology, because an n-dimensional real manifold had all its cohomology vanishing in degrees > n, but the group of order 2 seemed to have cohomology in infinitely many degrees and hence "must be infinite-dimensional". It took me a fair amount of time to get over this! In some sense I wish I'd learned group cohomology first... –  Kevin Buzzard Dec 12 '09 at 15:02
@kbuzz: The group of order 2 is infinite dimensional in the topological sense: its Eilenberg Mac Lane space is RP^{oo}. But I agree. I learned a little bit of group cohomology first in a finite group theory course (it was used -- only -- to prove the Schur-Zassenhaus Theorem) and it seemed quite bizarre compared to the singular cohomology that I by then knew pretty well. –  Pete L. Clark Dec 12 '09 at 16:53
@Pete: yes, I know that now. But my confusion was before a pertinent conversation with DC in 1996 :-) –  Kevin Buzzard Dec 12 '09 at 21:04
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"Do people just remember all the rules and go through the formal manipulations of the cohomology groups of class field theory mechanically, or are people actually "feeling" what is going on here. If it is the latter"

Let me be one to, if not advocate, at least defend the formal manipulation point of view. Perhaps ironically, given this, this will be an extremely hand-wavy response: Cohomology problems emanate from failure of exactness of a functor (i.e., "something you want to do to something else"), e.g., $A\rightarrow A^G$ for a $G$-module $A$. So you start with a short exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$, apply the "fixed by G" functor, and lo and behold, instead of a 0 on the right, you need a $H^1(G,A)$. Forgetting completely about what this object means/represents/is, it is the thing that is standing in the way of exactness. Then you go and notice that Hilbert's Theorem 90, or some other amazing result, tells you that the $H^1$ that you just ran into is zero. Voila! Exactness.

Now. Sometimes you find that $H^1$'s aren't always zero -- sad but true. So you start studying $H^1$'s in their own right. Some times these are manageable, or have been previously calculated, and some times not. Then you find, by looking at the long exact sequences in cohomology, that you could figure out an $H^1$ you need to know by looking at an $H^2$ (maybe you want an $H^1(G,C)$ given an $H^1(G,B)$ and an $H^2(G,A)$). And then lo and behold -- $H^2(G,A)$ happens to be a Brauer group or something else well-studied. Knowing this $H^2(G,A)$ trickles down to give you newfound knowledge of $H^1(G,C)$, which in turn gives you information about some failure of exactness on $H^0$'s (say, via some new short exact sequence $0\rightarrow C\rightarrow D\rightarrow E\rightarrow 0$ for which one would want $H^1(G,C)$), which is what you were trying to understand in the first place ("The house that Jack built" comes to mind). All this not having any idea what $H^2$ is!

And the process doesn't stop there -- $H^3$'s help you control $H^2$, which in turn control $H^1$'s, etc. I've never run into an $H^4$ in the wild, but they're not that scary for exactly this reason. They're just the thing standing in the way of an $H^3$ computation, and fit into the same exact sequences everything else does -- one leg at a time.

In any case, my point is not that you shouldn't try to understand cohomology groups on an intuitive level. It's that you shouldn't wait for a complete understanding of cohomology groups before you play around with the theorems to see what these groups are good for. The two should be learned in tandem, maybe even with a preference going toward being able to use them over being able to intuitively understand them.

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Von Neumann's quote comes to mind: “In mathematics you don't understand things. You just get used to them.” –  Dror Speiser Jan 29 '10 at 23:56
Or D'Alambert, talking to his students about calculus: "Allez en avant, et la foi vous viendra", which may be translated as "Push on and faith will catch up with you". :-) –  a.r. Jan 30 '10 at 4:23
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I'm not sure why none of the wise suggestions above got turned into an answer; my answer is pretty similar in content. My recommendation: learn homological algebra. Properly. Take a course on it, or seriously curl up with a book. The book I know best is Weibel's Homological Algebra. If you finish them and feel ready for more, you can try Gelfand and Manin (do NOT read these in the other order), or maybe somebody can leave other suggestions in comments.

Learning algebraic geometry is probably also a good idea, and recommended for any modern number theorist.

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I'd seriously recommend Mazur's "notes on etale cohomology of number fields" survey for a nice balance between formal manipulations and geometric intuition. Also at a more basic level, try Neukrich's "Algebraic number theory" for arithmetic-as-geometry philosophy.

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Mazur's notes are available online: numdam.org/item?id=ASENS_1973_4_6_4_521_0 –  Ben Linowitz Dec 13 '09 at 0:57
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