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Notes on class field theory

Kiran S. Kedlaya

Section 5.1 The setup of abstract class field theory

Reference.

[38], IV.4-IV.6. Remember that Neukirch’s cohomology groups are all Tate groups, so he doesn’t put the subscript “T” on them.

Subsection Abstract multiplicative groups and the class field axiom

We first introduce an abstract analogue of the groups \(K^*\text{,}\) for \(K\) a finite extension of \(k\text{,}\) and the norm maps between them. This enables us to state a key cohomological assumption.

Definition 5.1.1.

Let \(k\) be a field, let \(\kbar\) be an algebraic extension of \(k\text{,}\) and put \(G := \Gal(\kbar/k)\text{.}\) Let \(A\) be a \(G\)-module; for any subextension \(K\) of \(\kbar/k\text{,}\) define \(A_K = A^{\Gal(\kbar/K)}\text{.}\) (In the example where \(k\) is a local field, we will take \(\kbar\) to be the separable closure and \(A = \kbar^*\text{.}\))

Remark 5.1.2.

In this discussion, we are not going to make any explicit use of the field \(k\text{;}\) we are really just working with the profinite group \(G\text{.}\) One could extend this discussion to a general profinite group \(G\text{,}\) as is done in [38], by “pretending” that the profinite group corresponds to a field and its extensions via the Galois correspondence. That is, a “field extension” of \(k\) corresponds to a closed subgroup of \(G\text{;}\) a “finite extension” of \(k\) corresponds to an open subgroup of \(G\text{;}\) and so on.

Definition 5.1.3.

For \(L/K\) a finite extension of subextensions of \(\kbar/k\text{,}\) define the norm map \(\Norm_{L/K}\colon A_L \to A_K\) by \(\Norm_{L/K}(a) = \prod_g a^g\text{,}\) where \(g\) runs over a set of right coset representatives of \(G_L\) in \(G_K\text{.}\) In the Galois case this coincides with the norm map used in the definition of the Tate cohomology groups, except that we are using multiplicative notation rather than additive notation.
For \(L/K\) an infinite extension of subextensions of \(\)\(\kbar/k\text{,}\) we don’t have a well-defined norm map from \(A_L\) to \(A_K\text{.}\) By convention, however, we still write \(\Norm_{L/K} A_L\) to mean the intersection of \(\Norm_{M/K} A_M\) over all finite subextensions \(M\) of \(L/K\text{.}\)

Definition 5.1.4.

Set notation as in Definition 5.1.1. We say that \(A\) satisfies the class field axiom if for every cyclic extension \(L/K\) of finite subextensions of \(\kbar/k\text{,}\)
\begin{equation*} \#H^i_T(\Gal(L/K), A_L) = \begin{cases} [L:K] \amp i=0 \\ 1 \amp i = -1. \end{cases} \end{equation*}
Note that in general, it is not enough to impose this condition when \(K = k\text{.}\)
Since \(L/K\) is cyclic here, Theorem 3.4.1 implies that the groups \(H^i_T(\Gal(L/K), A_L)\) repeat with period 2. It will sometimes be convenient to work with \(i=1\) instead of \(i=-1\text{,}\) or with \(i=2\) instead of \(i=0\text{,}\) in order to have cohomology groups instead of Tate groups.
Under the class field axiom and the other conditions of abstract class field theory, for each finite Galois extension \(L/K\) of finite subextensions of \(\kbar/k\text{,}\) we will define a canonical isomorphism
\begin{equation*} r_{L/K}\colon \Gal(L/K)^{\ab} \to A_K / \Norm_{L/K} A_L \end{equation*}
(Theorem 5.3.9), which will moreover satisfy some compatibilities as we vary the field extension (Proposition 5.2.10, Proposition 5.2.10). Since we’ve already checked the class field axiom in the example where \(k\) is a local field and \(A = \kbar^*\text{,}\) this will recover the local reciprocity law (Theorem 4.1.2).

Subsection Abstract ramification theory

We next encode the key aspects of ramification theory into an abstract framework. At the moment this has nothing to do with the abstract units; the relationship will be made when we introduce abstract valuations a bit later.

Definition 5.1.5.

With notation as in Definition 5.1.1, let \(d\colon G \to \widehat{\ZZ}\) be a continuous surjective homomorphism. The example we have in mind is when \(k\) is a local field and \(d\) is the surjection of \(G\) onto \(\Gal(k^{\unr}/k) \cong \widehat{\ZZ}\text{.}\)
Define the Weil group of \(k\) as the subgroup \(d^{-1}(\ZZ)\) of \(G\text{.}\) This group will play an important role in the construction of the reciprocity map (see Definition 5.2.1).
We now set some more notation to mimic the example case.

Definition 5.1.6.

Define the inertia group \(I_k\) to be the kernel of \(d\text{.}\) Define the maximal unramified extension \(k^{\unr}\) of \(k\) to be the fixed field of \(I_k\text{.}\) Likewise, for any subextension \(K\) of \(\kbar/k\text{,}\) put \(I_K = G_K \cap I_k\) and let \(K^{\unr} = k^{\unr} K\) be the fixed field of \(I_K\text{.}\)
We say an extension \(L/K\) of subextensions of \(\kbar/k\) is unramified if \(L \subseteq K^{\unr}\text{.}\) This implies that \(G_L\) contains \(I_K\text{,}\) necessarily as a normal subgroup, and that \(G_L/I_K \subseteq G_K/I_K\) injects via \(d\) into \(\widehat{\ZZ}\text{.}\) Hence \(G_L/I_K\) is abelian and any finite quotient of it is cyclic; in particular, \(G_K\) is normal in \(G_L\) and \(\Gal(L/K) = G_L/G_K\) is cyclic. (Note also that \(K^{\unr}\) is the compositum of \(K\) and \(k^{\unr}\text{;}\) see Example 5.1.9.)
If \(K \neq k\text{,}\) then \(d\) need not map \(G_K\) onto \(\widehat{\ZZ}\text{,}\) so it will be convenient to renormalize things.

Definition 5.1.7.

Put
\begin{equation*} d_K := \frac{1}{[\widehat{\ZZ}: d(G_K)]}d\text{;} \end{equation*}
then \(d_K\colon G_K \to \widehat{\ZZ}\) is surjective and induces an isomorphism \(\Gal(K^{\unr}/K) \cong \widehat{\ZZ}\text{.}\)
Given a finite extension \(L/K\) of subextensions of \(\kbar/k\text{,}\) define the inertia degree (or residue field degree)
\begin{equation*} f_{L/K} := [d(G_K):d(G_L)] \end{equation*}
and the ramification degree
\begin{equation*} e_{L/K} := [I_K:I_L]. \end{equation*}
By design we have multiplicativity:
\begin{equation*} e_{M/K} = e_{M/L}e_{L/K}, \qquad f_{M/K} = f_{M/L}f_{L/K}\text{.} \end{equation*}
If \(L/K\) is Galois, we have an exact sequence
\begin{equation*} 1 \to I_K/I_L \to \Gal(L/K) \to d(G_K)/d(G_L) \to 1\text{,} \end{equation*}
from which we deduce the “fundamental identity”:
\begin{equation*} e_{L/K}f_{L/K} = [L:K]\text{.} \end{equation*}
The fundamental identity also holds if \(L/K\) is not Galois: let \(M\) be a Galois extension of \(K\) containing \(L\text{,}\) then apply the fundamental identity to \(M/L\) and \(M/K\) and use multiplicativity.

Subsection Abstract valuation theory

We next introduce an abstract analogue of the valuation maps on unit groups, which tie \(d\) and \(A\) together in a crucial way. The catch is that these valuations will be valued not in \(\ZZ\) but only in \(\widehat{\ZZ}\text{;}\) however, this is okay because we only need them to normalize the definition of the reciprocity map.

Definition 5.1.8.

With notation as in Definition 5.1.1 and Definition 5.1.5, a henselian valuation of \(A_k\) with respect to \(d\) is a homomorphism \(v\colon A_k \to \widehat{\ZZ}\) such that:
  1. the group \(Z = \im(v)\) contains \(\ZZ\) and satisfies \(Z/nZ \cong \ZZ/n\ZZ\) for all positive integers \(n\text{;}\)
  2. for every finite extension \(K\) of \(k\text{,}\) \(v(\Norm_{K/k} A_K) = f_{K/k} Z\text{.}\)
In this case, for each finite subextension \(K\) of \(\kbar/k\text{,}\) we obtain a henselian valuation \(v_K\colon A_K \to Z\) by setting
\begin{equation*} v_K = \frac{1}{f_{K/k}} \Norm_{K/k}\text{.} \end{equation*}
Then \(v_{K}(a) = v_{K^g}(a^g)\) for any \(a \in A\) and \(g \in G\text{,}\) and for \(L/K\) a finite extension of finite subextensions of \(\kbar/k\text{,}\) \(v_K(\Norm_{L/K}(a)) = f_{L/K} v_L(a)\) for any \(a \in A_L\text{.}\)

Example 5.1.9.

By design, the previous conditions are satisfied in the case where:
  • \(k\) is a local field of characteristic 0 and \(\kbar\) is its algebraic closure;
  • \(A = \kbar^*\) (the class field axiom is confirmed by Lemma 1.2.3 for \(H^1_T\) and Proposition 4.2.11 for \(H^0_T\));
  • \(d\colon \Gal(\kbar/k) \to \widehat{\ZZ}\) the map coming from the identification of \(\Gal(k^{\unr}/k)\) with \(\widehat{\ZZ}\text{;}\)
  • \(v\colon A_k \to \widehat{\ZZ}\) is the composition of the valuation \(k^* \to \ZZ\) with the inclusion \(\ZZ \to \widehat{\ZZ}\text{.}\)
One piece of content in this statement is the assertion that for any finite extension \(K\) of \(k\text{,}\) \(K^{\unr} \subseteq K k^{\unr}\text{.}\) This holds because for any finite unramified extension \(L\) of \(K\text{,}\) we can write \(L = K L_0\) where \(L_0\) is the unramified extension of \(k\) with the same residue field as \(L\text{.}\)

Example 5.1.10.

A basic example of these constructions occurs for \(k\) finite; see Exercise 1. A closely related example can be obtained from Example 5.1.9 by replacing the algebraic closure of \(k\) with its maximal unramified subextension.

Remark 5.1.11.

Suppose that we have an instance of Definition 5.1.8. Then for any \(c \in \widehat{\ZZ}^*\text{,}\) the map \(v\) is also a henselian valuation of \(A_k\) with respect to \(cd\text{,}\) but the definition of the reciprocity map will be affected; see Exercise 3.
Now suppose further that \(cZ = Z\text{.}\) Then \(cv\) is also a henselian valuation of \(A_k\) with respect to \(cd\text{,}\) and in this case the definition of the reciprocity map will be unaffected; see Exercise 4.

Subsection Cohomology of units

Before defining the reciprocity map, we collect some direct consequences of the class field axiom. These are very similar to arguments we used in the proof of local reciprocity except that there, we used the cohomology of unramified extensions to establish the class field axiom, whereas here we are moving information in the opposite direction!

Definition 5.1.13.

For any finite subextension \(K\) of \(\kbar/k\text{,}\) define the unit subgroup \(U_K\) as the set of \(u \in A_K\) with \(v_K(u) = 0\text{;}\) this definition extends naturally to infinite subextensions. We say that \(\pi \in A_K\) is a uniformizer for \(K\) if \(v_K(\pi) = 1\text{.}\)

Proof.

We’ll drop \(\Gal(L/K)\) from the notation, because it’s the same group throughout the proof. Keep in mind that an unramified extension is always Galois and cyclic, so we can apply periodicity of Tate groups (Theorem 3.4.1) and Herbrand quotients.
Consider the short exact sequence
\begin{equation*} 0 \to U_L \to A_L \to A_L/U_L \to 0\text{.} \end{equation*}
In this sequence, \(A_L/U_L\) is isomorphic to \(Z = \im(v)\) with trivial group action, so \(H^0_T(Z) = Z/\Norm(Z)\) is cyclic of order \([L:K]\) generated by \(\pi_L\) and \(H^{-1}_T(Z) = \ker(\Norm)\) is trivial. The long exact sequence in Tate groups looks like
\begin{gather*} 1 = H^{-1}_T(A_L/U_L) \to H^{0}_T(U_L) \to H^{0}_T(A_L) \\ \to H^{0}_T(A_L/U_L) \to H^{1}_T(U_L) \to H^1_T(A_L) = 1 \end{gather*}
and by the long exact sequence, the two ends of the middle arrow \(H^0_T(A_L) \to H^0_T(A_L/U_L)\) have the same order. It is thus enough to show that \(H^1_T(U_L) = 1\text{,}\) as then this map will be an isomorphism.
Here is where we use that \(L/K\) is unramified, not just cyclic: any uniformer of \(K\) is also a uniformizer of \(L\text{,}\) which allows us to split the surjection \(A_L \to A_L/U_L\) of \(\Gal(L/K)\)-modules (as in Corollary 4.2.6). This in turn means that \(H^1_T(U_L)\) is a direct summand of \(H^1_T(A_L)\text{,}\) and the latter vanishes by the class field axiom.

Proof.

Apply the \(H^0_T\) case of Proposition 5.1.14 to each finite subextension of \(L/K\text{.}\)

Exercises Exercises

1.

Show that the hypotheses of abstract class field theory (i.e., the class field axiom and the conditions on a henselian valuation) are satisfied in the following case:
  • \(k\) is a finite field and \(\kbar\) is its algebraic closure;
  • \(d\colon \Gal(\overline{k}/k) \to \widehat{\ZZ}\) is the usual isomorphism;
  • \(A\) is the group \(\ZZ\) with the trivial action;
  • \(v\colon A_k \to \widehat{\ZZ}\) is the inclusion of \(\ZZ\) into its profinite completion.
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