There is an idea (I think originally due to Tate) that class field theory is fundamentally a consequence of Pontrjagin duality and Hilbert Theorem 90. I'm curious whether this can phrased using modern (slightly motivic) language in something like the following 3-line proof sketch of global class field theory (the set-up is more than 3 lines :) ). The nub of the argument is a global compactly-supported Pontryagin duality statement #(5) below, which is standard in the function field case and which I wish (but don't know) to be true for global fields (and which is the real reason for this question).

Let $K$ be a number field with ring of integers $O$. For any collection of places $S$ of $K$ including all infinite ones, let's make believe that there is a "Galois cohomology" dg category (or maybe a stable infinity category) $\mathcal{A}$ which has as objects (at least) "complexes of abelian group schemes" over $Spec(O_S)$. Let's assume that this category satisfies the following properties.

(1) For any commutative algebraic group $A$ over $O_S$ there is an object, $A\in \mathcal{A}$.

(2) For any (reasonable) map of schemes $T\to Spec(S)$ and any $A\in \mathcal{A}$, we have a well-behaved (derived) stalk functor $\Gamma(T, A)$, with value the etale (or something) derived sections $T\to A$. Further, say we can evaluate the stalk of $A$ at $K_v$ for any place and $O_v$ for any place not in $s$. When $v$ is an infinite place, we set the derived stalk functor at $K_v$ to be the complex of (conjugation invariants in) chains on the corresponding Lie group.

(3) There is a constant motive, $const$, with $\text{RHom}(const, A) = R\Gamma(O_S, A)$ (derived sections of $A$ over $O_S$ in a suitable topology).

(4) The object $\mathbb{G}_m\in \mathcal{A}$ satisfies Hilbert Theorem 90, i.e. $R\Gamma^1 (Spec(K), \mathbb{G}_m) = 0$.

(5) The category admits an inner $\underline{Hom}$ and a global Pontryagin-Serre duality statement as follows.

For any object $A\in \mathcal{A}$, define its compactly supported derived sections $R\Gamma_c(O_S, A)$ to be the fiber of the map of complexes $R\Gamma(O_S, A)\to \prod_{v\in S} R\Gamma(K_v, A)$. Then Pontryagin duality states that $R\Gamma(A)$ and $R\Gamma(\underline{Hom}(A, \mathbb{G}_m))$ are dual complexes. (Note that infinite places are important here).

With this setup, we would have the following three-line proof of class field theory. Let $CFT_S$ be the group of characters of the Galois group unramified outside of $S$. Now we have (assuming $S$ contains all infinite places):

$CFT_S = R\Gamma^0(\text{Spec}(O_K), B(\mathbb{Q}/\mathbb{Z}))$ (as we are classifying Galois characters into $\mathbb{Q}/\mathbb{Z}$). Using standard derived machinery, we can take the $B\mathbb{Q}/\mathbb{Z}$ outside (with a shift) to get $CFT_S = H^1(R\Gamma(const)\otimes \mathbb{Q}/\mathbb{Z})$ (where the tensor product is taken in a suitably derived sense).

But now by duality, we have $R\Gamma(const) = R\Gamma_c(\mathbb{G}_m)^\vee$, which means that we are looking for $RHom^1(R\Gamma_c(\mathbb{G}_m), \mathbb{Q}/\mathbb{Z})$. Since $\mathbb{Q}/\mathbb{Z}$ is divisible (hence injective), we can take it outside, and write $CFT_S = Hom(R\Gamma_c^1(\text{Spec}(O_S), const), \mathbb{Q}/\mathbb{Z}).$

The definition of compact support now tells us that $H^1R\Gamma_c(\mathbb{G}_m)$ fits into an exact sequence $$H^0R\Gamma(O_K,\mathbb{G}_m)\to \prod_{v\in S} R\Gamma(K_v, \mathbb{G}_m)\to {\bf H^1 R\Gamma_c}\to H^1(O_S, \mathbb{G}_m).$$ Passing to the limit as $S$ becomes all places and using Hilbert's Theorem 90 to get rid of the $H^1$ on the right then gives us the usual statement of class field theory. $\square$

The one piece of the setup that seems to not be standard is a global duality with compact supports statement (5) above. There is a well-known analogue in the function-field case, and Dustin Clausen's thesis seems to rely on similar ideas, but I've never seen a mixed-characteristic statement of this sort in the literature (as far as I know, it may be false in general, or require a more homotopical language). Does anyone know whether something like this is true?

  • 2
    $\begingroup$ Can you, please, explicitly formulate "the usual statement of the global class field theory" that your argument yields? $\endgroup$ Sep 10, 2016 at 15:32
  • 2
    $\begingroup$ Sure. It's that a Galois character is equivalent to a map from $\prod_{v} K_v^\times$ to $\mathbb{Q}/\mathbb{Z}$ which vanishes on $K^\times.$ $\endgroup$ Sep 10, 2016 at 20:02
  • $\begingroup$ (with restricted product above). As maps to $\mathbb{Q}/\mathbb{Z}$ is an exact contravariant functor and we're hoping $H^0(K, \mathbb{G}_m)$ is Galois-invariant points of $\bar{K}^\times$ (or $\pi_0 K^\times$ in the infinite place case), the exactness of the sequence above would imply this CFT statement. $\endgroup$ Sep 10, 2016 at 20:09
  • 9
    $\begingroup$ I'm not a native speaker of this derived homotopical language, but I fear that your exact sequence may have got slightly mangled (shouldn't the second term be an $H^1$?) Anyway, if you're happy to localise at a prime (which is going to be sufficient if you just want to classify finite-order characters of Galois groups) then all the formalism you describe is standard in the theory of Galois representations (see e.g. Nekovar's "Selmer complexes"). I have grave doubts about whether you can set up this machinery without having already proved global CFT along the way, though. $\endgroup$ Sep 12, 2016 at 8:15
  • 1
    $\begingroup$ @DavidLoeffler Thanks! Localizing the coefficient field at a prime is certainly ok. I'm not very literate in Galois representations formalism, but the Poitou-Tate result in Nekovar's notes (0.7.1) looks very applicable. Is this what you meant? $\endgroup$ Sep 12, 2016 at 23:12

1 Answer 1


This is a late answer, but I would recommend that anyone interested in Dmitry's heuristic argument read "Notes on etale cohomology of number fields," by Barry Mazur.

Mazur's article is an exposition of Artin-Verdier's duality theorem, which is a version of Dmitry's (5). (Artin-Verdier duality says that $\mathbb G_m[3]$ is a dualizing complex for constructible sheaves-- there should be a shift by $3$ in Dmitry's statement and the first complex should be $\rm R \Gamma_c(A)$). He explains how the content of class field theory can be read out from the duality theorem and the computation of the cohomology of $\mathbb G_m$.

Of course Mazur emphasizes that the main input into the proof of the duality theorem (and the computation of the cohomology of $\mathbb G_m$) is class field theory. So there is no proof that avoids circularity, but the resulting conceptual picture is very nice.

  • $\begingroup$ Well, in a remark (on page 538) Mazur asks about a possibility to compute $H^*(-;\mathbb G_m)$ directly for global fields, as in the function field case. $\endgroup$ Apr 27 at 4:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.