What's the use of group cohomology for class field theory?

Question

I'm a graduate student studying now for the first time class field theory.
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.
For example here Learning Class Field Theory: Local or Global First? was discussed the local vs. global approach, here How many ways are there to teach class field theory? were listed several difference references and ways to learn about the subject, and the list could go on and on.

This is all to say that I am aware that this may seem like a duplicate, but my question is of a slightly different character.

I am following mainly the cohomological approach to learn about the subject, which I've taken to be important since so many experts here and elsewhere value it as such. It is however clear to that to obtain all the theorems one need not all the machinery of finite groups cohomology, which I have no doubt is important to learn.

My Question then is: Why is it of such importance to learn about (finite) group cohomology?

In particular:

• Is it a necessary instrument for the study of further branches of (algebraic and geometric) number theory?
• Is it necessary to do the "calculations" in further graduate subjects?
• Does it provide insight or is it the base case of any further instrument?
• Which particular insight does it provide in class Field Theory to make it such a relevant approach?

The purpose of the four stated points is to specify what I mean by "use" and what would be an optimal answer. Naturally, since I am still quite ignorant, if the main "use" of it is something completely else and this will be pointed out to me in the comments, I shall change the edit accordingly.

Thank you in advance for any reply.

Answer 1

First of all, as already said by others: Classical class field theory can be formulated entirely without cohomology, so it is a choice to use it.

Benefits of using group cohomology:

If you use Galois cohomology, the main theorems of class field theory can be phrased as statements looking a lot like a form of Poincaré Duality, e.g. local class field theory becomes a nondegenerate duality

$H^i(F,M) \otimes H^{2-i}(F,M^\ast) \rightarrow \mathbb{Q}/\mathbb{Z}$

which looks a bit as if your local field was some kind of 2-dimensional compact manifold (except that the top cohomology group is $\mathbb{Q}/\mathbb{Z}$ instead of $\mathbb{Z}$.

There is a similar reformulation of global class field theory (e.g., look for Artin-Verdier duality).

So, if you like topology, one benefit of choosing to use group cohomology is that you can try to get inspiration/intuition from topology.

Then the Galois group is a little like the fundamental group of "that manifold" and indeed some explicit presentations of the Galois group of a local field look [very vaguely...] a little like presentations of fundamental groups of surfaces. This is a vague analogy at best, but analogies often help us to find things "more natural".

As a side note: Etale cohomology also has a Poincaré Duality theorem, but even though Galois cohomology can be regarded as a special case of étale cohomology, the Poincaré Duality Theorem of étale cohomology says nothing about class field theory. These dualities are disjoint phenomena.

Second: Many things need not be proven afresh. For example, there are maps between group cohomology groups when you go to a smaller or bigger group (restriction and corestriction). From the field perspective this corresponds to going to a bigger field or subfield. If you develop group cohomology, there are automatically induced maps on all group cohomology (or homology) groups. If you avoid group cohomology and instead work with things like the Brauer group, defined through central simple algebras, the respective maps need to be set up manually and you all the time need to verify a lot of little properties which would be automatic if you simply imported them from abstract homological algebra.

Third: Linking back to the first: At some point in your life you might wish to combine number theory with geometry, and consider varieties over number fields. For the varieties you will probably use some cohomology theories of geometric flavour (e.g., etale cohomology of the variety base changed to the separable closure of the base field), or more arithmetic invariants like Chow groups, etc. These invariants call all be phrased cohomologically. For this reason, if you want to combine them with number theory, you benefit a lot if you have also phrased your class field theory in terms of cohomology because then all these concepts "mix in a friendly compatible way". If you avoid using cohomology, you would have to build a lot of connecting bridges beforehand.

Some invariants of arithmetic relevance, e.g., motivic cohomology groups, have been modelled really with topology inspiration in mind (If you read Voevodsky, he clearly thought mostly in terms of homotopy theory). Yet, although topologically inspired they then turned out to be extremely valuable in number theory, too. The same is true for $K$-theory. Many of these things, if one hides their group cohomology origin, might at first appear extemely unmotivated and enigmatic.

Answer 2

Galois cohomology, which is a special case of group cohomology, is used pretty heavily in many areas of algebraic number theory: especially in Galois deformations and the Langlands program, but also in Iwasawa theory and other areas.

It's often necessary to do calculations with Galois cohomology in graduate-level learning in these other areas, though it may have a somewhat different flavor from the calculations used in class field theory.

Answer 3

This is not a full answer, but is a bit too long for a comment. I recommend the introduction to the second part of Lang's Algebraic Number Theory, where he discusses several different approaches to class field theory, and says that ‘no one piece of insight which has been evolved since the beginning of the subject has ever been "superseded" by subsequent pieces of insight.’ In my opinion, class field theory remains a miracle even after you've worked through one (or more!) of the proofs, and the different proofs are just highlighting different aspects of the miraculous structure involved. Lang notes that the cohomological approach is a relative latecomer; the main theorems of class field theory were in place by 1927, but it wasn't until about 1950 that Hochschild introduced cohomology to show that simple algebras, which formed the basis of an earlier approach to class field theory due to Hasse, could be largely bypassed.

More generally, I would say that whenever you find cohomology (group cohomology or otherwise) lurking in the structure you're studying, it's worth trying to understand it, because it frequently yields some nontrivial conclusions almost "for free"—or at least it suggests certain obvious further questions to pursue. In number theory specifically, Lang makes the following comment (but I should say that I don't know exactly what he's alluding to; maybe someone with more expertise in this area than I have can provide further references):

This [cohomological] approach, which shows that the second cohomology group of the idele classes (for the algebraic closure) is isomorphic to $\mathbb{Q}/\mathbb{Z}$, provides a good background for theories where this result is used to obtain pairings, e.g., some diophantine questions related to abelian varieties over $p$-adic fields or number fields as in the work of Tate.

Answer 4

In addition to other good answers: I'd tend to recommend taking a cohomological approach to classfield theory as a second pass through the subject, so that you already know the down-to-the-metal number-theoretic facts, and can focus on re-interpreting them in (co-)homological terms. Or, oppositely, if one already is well acquainted with cohomological things, then it's "just" seeing how they may apply to classfield theory. "One thing at a time" is often best, in my own experience.

For practice, be sure you can rewrite Hilbert's theorem 90 homologically. EDIT: per LSpice's apt comment, it would be most accurate to say "cohomologically" here, just so no one is inadvertently misled. :)

I'd also strongly recommend not taking the most traditional approach to group cohomology (with "(in)homogeneous bar resolutions" as the definition) ... as opposed to a more modern (1950s!?!) approach via derived functors of very natural functors, namely, the (co-)fixed-vector functors $M\to M^G$ and $M\to M_G$. This shows the commonality with sheaf cohomology, Lie-group (co)homology, Lie-algebra (co)homology, $(\mathfrak g,K)$-cohomology and lots of other similar things. Yes, for computations, sometimes we do want explicit (small) projective or injective resolutions, but it's not optimal to take those as the definition... especially since the whole general machine is well established.

In particular, some of the colorfully-named homological mechanisms in classfield theory are really special cases of far more general homological phenomena. Herbrand quotient. Dimension shifting. In particular, to imagine a necessity of seeing the number-theoretic content in these mostly homological ideas is misguided. At the same time, insufficient homological chops can leave a person "stranded" doing ugly approximations thereof, thinking that it's supposed to be "number theory". :)

One more EDIT: it has always amazed me that in local or global classfield theory, the "reciprocity law" map is a cup product in cohomology!!! With a sort of "fundamental class"... and so on. But I'm glad I already knew about quadratic reciprocity, Kronecker-Weber, and so on... :)