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I am currently learning (local) class field theory via group cohomology with Milne's notes. I have a number of questions about using group cohomology to prove the main statements of class field theory. The first thing: how do you motivate the appearance of group cohomology in class field theory? The best thing I could find is: the group cohomology groups $H^r(G,M)$ are the same as the singular cohomology of an aspherical space with coefficients in $M$. (When the space is aspherical, the homology groups only depend on the space having $G$ as its fundamental group). Anyway, now for class field theory, somehow there is singular homology (and the topology of the classifying space?) in the background that is giving us these results. I think if I can understand this more closely then I can be more comfortable with the machinery being used. Does there exist references that develop class field theory with interpretations closely related to the "background singular homology"? How did people even come up with the cohomological proofs, did they have these ideas?

Thank you

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    $\begingroup$ You can develop group cohomology without saying anything about spaces; it's a purely algebraic operation and, in this context, should probably be thought of in that way. $\endgroup$ Jul 6, 2014 at 3:24
  • $\begingroup$ thanks for the reply. I understand that group cohomology is now completely algebraic (as is given in Milne's notes). But if it is not explained where all these seemingly strange definitions come from, doesn't it seem a little ad hoc? $\endgroup$
    – Elliot
    Jul 6, 2014 at 3:38
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    $\begingroup$ My point is thinking in terms of aspherical spaces is likely to be unhelpful, even if this was an important part of the historical development. There are reasons to invent cohomology that don't directly have anything to do with spaces, and those reasons are likely to be the ones relevant to class field theory (as in the answer below). $\endgroup$ Jul 6, 2014 at 3:40
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    $\begingroup$ The short, but not very helpful answer, is that cohomology should be thought of as an aspect of higher category theory, and higher categories naturally show up in many places, including number theory and algebra, not just in algebraic topology. But this perspective is not very helpful for a beginner. For the sake of completeness, ncatlab.org/nlab/show/cohomology in general and ncatlab.org/nlab/show/Brauer+group for the higher-categorical interpretation of the Brauer group in particular. $\endgroup$ Jul 6, 2014 at 3:48
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    $\begingroup$ (I could be more specific if I actually knew why cohomology shows up in class field theory, but unfortunately I don't. Maybe the simplest advice is not to worry about it for now; things will hopefully become clearer with experience.) $\endgroup$ Jul 6, 2014 at 3:49

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Tate writes in his article in Cassels and Frolich that the cohomology calculations have their origin in genus theory (going back --- at least --- to Gauss's Disquitiones).

I haven't gone that far back, but I have read parts of Hilbert's Zahlbericht, where e.g. his Theorem 90 appeared. (The theorems are labelled in order throughout the book.)

If you look at how Hilbert argues there, he uses Theorem 90 in much the same way we do --- it gives a vanishing result, which can be fed into other computations via the snake lemma (i.e. the boundary map in the cohomology long exact sequence). Indeed, he has many calculations that are concrete forms of the snake lemma in the book.

There are also computations with units, especially in the case of cyclic extensions, which are analogous to what we would describe as computing the Herbrand quotient of the unit group (which nowadays appears as an input along the way to computing the Herbrand quotient of the idele class group for a cyclic extension). (Note that these sorts of computations go back to Kummer. The chapter by Rosen in Modular Forms and Fermat's Last Theorem gives some nice insight into this.)

As Rene Schipperus notes in another answer, there was a (at least apparently) separate thread in the first half of the 20th century, in which Brauer, Hasse, and Noether were studying central simple algebras over number fields, and discovered and proved that the sum of the local invariants is zero. They then realized that this result could be applied to reprove Artin's reciprocity law. (See this historical survey for more details.)

When the theory of central simple algebras was recast cohomologically, this development was unified with the earlier arguments of genus theory, and the modern cohomological treatment emerged.

(There is yet another thread of development, which is the replacement of $L$-functions in the argument with algebraic arguments via Kummer theory, which isn't quite as relevant to your question.)

Genus theory isn't so commonly treated in modern alg. no. theory books. Advanced number theory, by Harvey Cohn, gives a discussion for quadratic fields (although the cohomological aspects are not made explicit, you can see that Hilbert's Thm. 90 plays an important role). Here is an exercise sheet that turns up when you google genus theory, and which is more cohomological.

[Incidentally, while group cohomology literally came out of alg. topology, the computations in alg. number theory that it captures are much older! E.g. Hilbert's Thm. 90, at least in the cyclic case, certainly predates the invention of group cohomology by algebraic topologists. One thing that group cohomology is good at is expressing the non-cylic case; my memory is that there was a period when people --- maybe Emmy Noether --- expressed the non-cyclic case of HT90 in the language of crossed homomorphisms etc. that can be used to talk about group $H^1$. It must have been a relief to discover that all the different computations, crossed homomorphisms, factor sets, local invariants, proofs of the Artin rec. law, and so on, could be unified by the language of gp. cohomology!]

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The cohomology formulation came at the end of a long line of development. Briefly, local class field theory was developed using Brauer groups, and Brauer groups can be interpreted as cohomology groups, then it was seen that one could dispense with the Brauer formalism and deal directly with the cohomology groups, finally the basic maps were interpreted as cap products. Jacobson's Basic Algebra II presents the basic theory of Brauer groups along with some number theory applications. The book by Pierce, Associative Algebra treats the subject in great detail. Also if you are really into the history the papers of Nakayama are nice.

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  • $\begingroup$ thank you for the reply. I have read that group cohomology comes from replacing the use of central simple algebras. Are there no interesting interpretations on the singular side of things? Or does group cohomology just come up because it's more efficient and convenient to phrase everything in this way. From what people tell me, it seems that this is the case $\endgroup$
    – Elliot
    Jul 6, 2014 at 3:41
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    $\begingroup$ Group cohomology originated in algebraic topology. It was a theorem of Hopf that if all the higher homotopy groups of a space vanish then the cohomology is determined by the fundamental group, it was Eilenberg and Maclane who showed we how to this determination worked, and on fact the cohomology groups of the space are the group cohomology of the fundamental group. $\endgroup$ Jul 6, 2014 at 4:09

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