Skip to main content

Notes on class field theory

Kiran S. Kedlaya

Section 7.6 Brauer groups and the reciprocity map

Reference.

[37] IV (for the general theory of Brauer groups); VII.7 and VII.8 (for the application to reciprocity). For the general theory, see also [42], 1.5 for a quick presentation or [24], Chapter 4 for something more thorough.
We discuss Brauer groups of fields, especially number fields. On one hand these can be used to give an alternate construction of the global reciprocity map, not based on abstract class field theory, by taking local-global compatibility as the source of the definition and then checking the reciprocity isomorphism using Tate’s theorem. On the other hand, Brauer groups carry important information from class field theory which is useful in numerous applications, such as finding rational points on algebraic varieties (Remark 7.6.21).
In this lecture, we reprise a bit of shorthand from Section 4.2, writing \(H^i(L/K)\) to mean \(H^i(\Gal(L/K), L^*)\text{.}\) Beware that [37] uses a different convention here, writing \(H^i(L/K)\) to mean \(H^i(\Gal(L/K), C_L)\text{.}\)

Subsection The Brauer group of a field

Definition 7.6.1.

Recall from Definition 4.1.15 that we have defined the Brauer group of a field \(K\) as the group
\begin{equation*} \Br(K) := H^2(\overline{K}/K) = \varinjlim_{L/K} H^2(L/K) \end{equation*}
where \(L\) runs over finite Galois extensions of \(K\) and the transition maps in the direct limit are inflation maps. By Lemma 1.2.3 and Proposition 4.2.13, these maps are all injective, so the direct limit is actually a union.
This definition of Brauer groups is not the original one; we give that next. The standard definition is not needed for the results of class field theory, but is important in other contexts such as Remark 7.6.21.

Definition 7.6.2.

An Azumaya algebra of rank \(n\) (a positive integer) over a field \(K\) is a (not necessarily commutative) \(K\)-algebra \(A\) such that \(A \otimes_K \overline{K}\) is isomorphic as a \(\overline{K}\)-algebra to the matrix algebra \(M_n(\overline{K})\text{.}\) One way this can occur is for \(A\) to itself be isomorphic to the matrix algebra \(M_n(K)\text{;}\) when this occurs we say that \(A\) is split.
There are various equivalent characterizations of Azumaya algebras ([42], Proposition 1.5.2). For example, \(A\) is an Azumaya algebra over \(K\) if and only if it is isomorphic to a matrix ring over some division algebra (skew field) \(D\) which is itself a finite dimensional \(K\)-algebra; and the isomorphism class of \(D\) is uniquely determined by that of \(A\text{.}\)

Proof.

See [37], Corollary IV.3.16; [48], Chapter X, Proposition 9; or [24], Theorem 8.11.

Example 7.6.4.

For \(K\) an algebraically closed field, Lemma 7.6.3 includes the assertion that every division algebra which is finite dimensional over \(K\) is trivial. We may check this directly as follows. If \(D\) is such a division algebra, then for each \(x \in D\text{,}\) multiplication by \(x\) defines a \(K\)-linear endomorphism of \(D\text{,}\) which necessarily has at least one eigenvalue \(y \in K\text{.}\) Then \(x-y\) is an element of \(D\) which cannot be invertible (since multiplication by this element is a \(K\)-linear endomorphism of \(D\) with \(0\) as an eigenvalue), so it must be zero; hence \(x \in K\text{.}\)

Example 7.6.5.

For \(K\) a finite field, \(\Br(K)\) is trivial. In the classical interpretation, this is Wedderburn’s theorem that every finite division algebra is commutative. In the cohomological interpretation, it follows from Proposition 4.2.4 via the periodicity of Tate groups (Theorem 3.4.1).

Remark 7.6.6.

The property of a field \(K\) of characteristic 0 having trivial Brauer group is useful in the theory of finite group representations: for such a field, any \(K\)-valued character of a finite group arises from a representation defined over \(K\text{.}\) (This follows from Schur’s lemma: the character in question appears within some irreducible \(K\)-linear representation, whose endomorphism ring is a division algebra; the triviality of the Brauer group forces this to split without any base extension.)
By contrast, for \(G = \{\pm 1, \pm i, \pm j, \pm k\}\) the unit quaternion group, the standard 2-dimensional representation of \(G\) has a \(\QQ\)-valued character but cannot be realized as a representation over \(\QQ\text{.}\) In other words, this representation has nontrivial Schur index.

Remark 7.6.7.

One can also associate Brauer groups to arbitrary rings and even to schemes in algebraic geometry, by extending the definition of Azumaya algebras suitably. One again has a relationship with Galois cohomology (or rather étale cohomology), but there is a bit more subtlety involved. See [15] for a classical reference or [42], section 6.6 for a modern treatment.

Subsection The Brauer group of a number field

We state the formula for the Brauer group of a number field, and prove it modulo one key step which we will cover a bit later. To accommodate readers who skipped over abstract class field theory, we present the argument in such a way as to avoid using global reciprocity. We will however use a weaker reciprocity statement for cyclotomic extensions whose proof does not require prior knowledge of the general case.

Proof.

The surjectivity is a direct consequence of Artin reciprocity for a cyclotomic extension (Definition 1.1.7). Alternatively, it can be deduced from the First Inequality (Proposition 7.3.7).
For \(K = \QQ\text{,}\) the fact that \(r_{L/K}\) factors through \(C_K\) follows from the explicit description of the local reciprocity maps given in Remark 6.4.8. In general, from Proposition 4.3.11 we have a commutative diagram
Figure 7.6.9.
and we know the bottom row kills principal idèles and the right column is injective. Thus the top row kills principal idèles too.

Proof.

The left square comes from applying Theorem 3.4.1 to the morphism \(L^* \to I_L\) of \(\Gal(L/K)\)-modules. Since \(r_{L/K}\) is defined in terms of local reciprocity maps, the right square comes from Lemma 4.2.20.
To deduce that both rows of the diagram are exact, it now suffices to check for the top row. We get an exact sequence
\begin{equation*} 0 \to H^0_T(\Gal(L/K), L^*) \to H^0_T(\Gal(L/K), I_L) \to H^0_T(\Gal(L/K), C_L) \to 0 \end{equation*}
by noting that \(H^{-1}_T(\Gal(L/K), C_L) = 0\) by Theorem 7.2.9 and \(H^1_T(\Gal(L/K), L^*) = 0\) by Lemma 1.2.3. Since \(L/K\) is cyclic, we may also apply Theorem 7.1.2 and Theorem 7.2.9 to see that \(\# H^0_T(\Gal(L/K),C_L) = n\text{.}\) Since \(L/K\) is cyclotomic, we may apply Proposition 7.6.8 to obtain a surjective map
\begin{equation*} C_K/\Norm_{L/K} C_L \to \Gal(L/K) \end{equation*}
between two groups of order \(n\text{;}\) it is thus an isomorphism and we are done.

Proof.

The map \(\Br(K) \to \bigoplus_v \Br(K_v)\) is the injection from Theorem 7.2.13. The value of \(\Br(K_v)\) for \(v\) finite is given by Lemma 4.2.20. For \(v\) complex, it is evident that \(\Br(K_v) = 0\text{.}\) For \(v\) real, by Theorem 3.4.1 we have
\begin{align*} \Br(\RR) &= H^2(\CC/\RR)\\ &\cong H^0_T(\CC/\RR)\\ &\cong \RR^* / \Norm_{\CC/\RR} \CC^* = \RR^*/\RR^+ \cong \tfrac{1}{2} \ZZ/\ZZ\text{.} \end{align*}
Since the values of \(\Br(K_v)\) are the ones given, the surjectivity of the map \(\bigoplus_v \Br(K_v) \to \QQ/\ZZ\) is evident. It thus remains to establish exactness at the middle of the sequence; that is, we must check that a class in \(\bigoplus_v \Br(K_v)\) maps to zero in \(\QQ/\ZZ\) if and only if it arises from some element of \(\Br(K)\text{.}\)
Fix such a class \((\alpha_v)_v \in \bigoplus_v \Br(K_v)\) and let \(S\) be the finite set of places \(v\) at which \(\alpha_v \neq 0\text{.}\) Let \(n\) be the least common multiple of the orders of the \(\alpha_v\text{.}\) Using Exercise 1, we can find a cyclic cyclotomic extension \(L/K\) such that for each \(v \in S\text{,}\) for some (hence any) place \(w\) of \(L\) over \(v\text{,}\) the degree \([L_w:K_v]\) is divisible by \(n\text{.}\) By Lemma 4.2.20, this means that \(\alpha_v \in H^2(L_w/K_v)\) for all \(v\text{.}\) We can now use Lemma 7.6.10 to argue in one direction: if \(\alpha\) maps to zero in \(\QQ/\ZZ\text{,}\) then it lifts to \(H^2(L/K) \subset \Br(K)\text{.}\)
To argue in the opposite direction, we need a bit more: we need to know that every class of \(\Br(K)\) belongs to \(H^2(L/K)\) for some cyclic cyclotomic extension \(L\) of \(K\text{.}\) For this, see Proposition 7.6.14.

Definition 7.6.13.

For \(K\) a number field and \(\alpha \in \Br(K)\text{,}\) the image of \(\alpha\) in \(\Br(K_v)\) is often called the local invariant of \(\alpha\) at \(v\text{.}\) The exact sequence appearing in Theorem 7.6.12 is sometimes called the fundamental exact sequence associated to \(K\text{;}\) it can be viewed as another source of “reciprocity” in class field theory. For example, applying the fundamental exact sequence to a quaternion algebra over \(\QQ\) (see Exercise 5) gives rise to Hilbert’s reformulation of the law of quadratic reciprocity using Hilbert symbols.
The fundamental exact sequence also plays a key role in various applications of Brauer groups in number theory. See for example Remark 7.6.21.

Subsection All Brauer classes are (cyclic) cyclotomic

Recall that in the cohomological approach to local class field theory, the crucial computation was that of the Brauer groups of local fields, which involved first studying unramified extensions and then transferring the knowledge to general extensions (see the proof of Proposition 4.2.17). The missing step in Theorem 7.6.12 is of a very similar nature, except that we have to vary the extension based on the class.

Proof.

As in the proof of Theorem 7.6.12, we can find a cyclic cyclotomic extension \(M\) of \(K\) such that for each place \(v\) of \(K\text{,}\) for some place \(w\) of \(M\) above \(v\text{,}\) the image of \(x\) in \(\Br(K_v)\) belongs to \(H^2(M_w/K_v)\text{;}\) we will prove the claim for this choice of \(M\text{.}\) By inflation-restriction (Proposition 4.2.13),
\begin{equation*} 0 \to H^2(M/K) \stackrel{\Inf}{\to} H^2(ML/K) \stackrel{\Res}{\to} H^2(ML/M) \end{equation*}
is exact; it thus suffices to check that the image of \(x\) via \(H^2(L/K) \stackrel{\Inf}{\to} H^2(ML/K)\) maps to zero in \(H^2(ML/M)\text{.}\) By Theorem 7.2.13 it suffices to check that we get the zero class in \(\Br(M_w)\) for each place \(w\) of \(M\text{,}\) but this follows from our choice of \(M\) as in the proof of Proposition 4.2.17.

Remark 7.6.15.

By Proposition 7.6.14, the field \(\QQ^{\ab}\) has trivial Brauer group. Since in addition every complex character of a finite group has values in \(\QQ^{\ab}\text{,}\) it follows that every irreducible complex representation of a finite group can be realized over \(\QQ^{\ab}\text{;}\) for a more direct proof of this, see [47], Chapter 12, Theorem 24.

Subsection Local-global compatibility via Brauer groups

We now turn things around and show that Proposition 7.6.14 can be used to recover local-global compatibility for the reciprocity map. More precisely, we show that the product of the local reciprocity maps is trivial on principal idèles (Proposition 6.4.7) and induces reciprocity isomorphisms (Remark 7.6.18) without the use of abstract class field theory, but with similar inputs (notably the First and Second Inequality).

Proof.

To begin with, Proposition 7.6.8 implies that \(r_{L/K}\) kills principal idèles whenever \(L/K\) is a cyclotomic extension, and Lemma 7.6.10 implies that in this case the composite \(H^2(L/K) \to \frac{1}{n} \ZZ/\ZZ\) along the bottom row of Figure 7.6.11 vanishes. By Proposition 7.6.14, the same then holds for any cyclic extension \(L/K\text{.}\) By Lemma 7.6.10 again, the composition along the top row of Figure 7.6.11 vanishes, proving the claim.

Remark 7.6.17.

Let \(L/K\) be a cyclic extension of number fields. At this point, \(r_{L/K}\) kills both principal idèles (by Proposition 6.4.7) and norms (since it does so locally), so it induces a map \(C_K/\Norm_{L/K} C_L \to \Gal(L/K)\text{.}\) By the surjectivity of the Artin map, as deduced from the First Inequality (Proposition 7.3.7), this map is surjective; by comparing orders using the Second Inequality (Theorem 7.2.9), we see that the map is also an isomorphism. This establishes local-global compatibility (Proposition 6.4.7) for cyclic extensions, from which it directly follows also for abelian extensions (using the structure theorem for finite abelian groups).
For general \(L/K\text{,}\) \(r_{L/K}\colon I_K \to \Gal(L/K)^{\ab}\) agrees with \(r_{M/K}\colon I_K \to \Gal(M/K)\) when \(M/K\) is the maximal abelian subextension of \(L/K\text{.}\) In particular, \(r_{L/K}\) once again vanishes on \(K^*\text{.}\) Hooray again!

Remark 7.6.18.

Note that for a cyclic extension \(L/K\) of number fields, Remark 7.6.17 establishes not just local-global compatbility, but the entire reciprocity isomorphism
\begin{equation*} C_K/\Norm_{L/K} C_L \cong \Gal(L/K)^{\ab} \end{equation*}
without use of abstract class field theory.
One can extend this conclusion to the case where \(L/K\) is abelian as follows. In this case, local reciprocity (Theorem 4.1.2) and Remark 7.6.17 together imply that we have a well-defined map. Using the cyclic case, we may see that this map is surjective; by Corollary 7.2.7 (a side effect of our proof of the Second Inequality), the map is forced to be an isomorphism.
For a general extension \(L/K\) with maximal abelian subextension \(M/K\text{,}\) we have from Remark 7.6.17 a well-defined map \(r_{L/K}\colon C_K \to \Gal(L/K)^{\ab}\) which coincides with \(r_{M/K}\colon C_K \to \Gal(M/K)\text{.}\) However, now we must be careful because the completion of \(L\) at some place might have maximal abelian subextension strictly larger than the corresponding completion of \(M\text{;}\) consequently, we cannot directly apply local norm limitation to deduce global norm limitation. In particular, we have an induced map \(r_{L/K} \colon C_K/\Norm_{L/K} C_L \to \Gal(L/K)^{\ab}\) but we currently only know that it is surjective. To resolve this, see Proposition 7.6.19 and Remark 7.6.20.

Subsection \(H^2\) and global reciprocity

As noted in Remark 7.6.18, we have a bit more work to do to establish the global reciprocity isomorphism without abstract class field theory: so far we only have it for abelian extensions, but we need the stronger version for arbitrary extensions to deduce the norm limitation theorem. We get this by calculating \(H^2(\Gal(L/K), C_L)\text{.}\)

Proof.

From the Second Inequality (Theorem 7.2.9) we have that \(\#H^2(\Gal(L/K), C_L) \leq [L:K]\text{,}\) so it suffices to produce a cyclic subgroup of order \([L:K]\) with a canonical generator. For this, note that by taking cohomology of the exact sequence
\begin{equation*} 0 \to L^* \to I_L \to C_L \to 0 \end{equation*}
and using the vanishing of \(H^1(\Gal(L/K), C_L)\) (Theorem 7.2.9 again), we obtain an exact sequence
\begin{equation*} 0 \to H^2(\Gal(L/K), L^*) \to H^2(\Gal(L/K), I_L) \to H^2(\Gal(L/K), C_L)\text{.} \end{equation*}
Using Theorem 7.6.12 we see that the cokernel of \(H^2(\Gal(L/K), L^*) \to H^2(\Gal(L/K), I_L)\) admits a canonical isomorphism with \(\frac{1}{[L:K]}\ZZ/\ZZ\text{,}\) proving the claim.

Remark 7.6.20.

At this point, we can construct a new map
\begin{equation*} \Gal(L/K)^{\ab} \cong H^{-2}_T(\Gal(L/K), \ZZ) \to H^0_T(\Gal(L/K), C_L) \cong C_K/\Norm_{L/K} C_L \end{equation*}
by taking the cup product with the canonical generator of \(H^2(\Gal(L/K), C_L)\) produced in Proposition 7.6.19. Since the hypotheses of Tate’s theorem (Theorem 4.3.4) are satisfied (the vanishing of \(H^1\) by Theorem 7.2.9, the value of \(H^2\) by Proposition 7.6.19) this map is an isomorphism.
Returning to Remark 7.6.18, we see that the map \(r_{L/K} \colon C_K/\Norm_{L/K} C_L \to \Gal(L/K)^{\ab}\) that we constructed from global reciprocity is a surjective map between two groups which are now known to be of the same order! We conclude that this map is an isomorphism, thus confirming Theorem 6.4.3 and Theorem 6.4.5. (Note that we avoided having to check that the isomorphism in the previous paragraph is compatible with the isomorphism coming from local-global compatibility.)
At this point, the reader who is avoiding abstract class field theory can now return to Section 7.4 for the proof of the existence theorem: the argument given therein (Theorem 7.4.8) depends only on the formulation of global reciprocity, not any particular method of proof.

Subsection An application to rational points on varieties

The computation of the Brauer groups of number fields has many interesting applications, including to the study of rational points on varieties. We remark on this point here.

Remark 7.6.21.

Let \(X\) be an algebraic variety over a number field \(K\text{.}\) For each place \(v\) of \(K\text{,}\) one can ask whether the set of \(K_v\)-rational points \(X(K_v)\) is empty. If this holds for some \(v\text{,}\) then evidently \(X(K)\) is also empty. However, the converse is generally not true; that is, there is no local-to-global principle for the existence of rational points on a general algebraic variety.
It was first observed by Manin that this failure can sometimes be overcome using the Brauer group \(\Br(X)\) (Remark 7.6.7). The key relevant point for now is that the construction of \(\Br(X)\) is contravariantly functorial, so any \(K\)-rational point \(x \in X(K)\) gives rise to a homomorphism \(\Br(X) \to \Br(K)\text{.}\) We can view this construction as a “pairing” \(X(K) \times \Br(X) \to \Br(K)\text{,}\) although it is only linear in the second argument as there is no natural group structure on \(X(K)\) in general.
For each place \(v\) of \(K\text{,}\) we can form the base extension \(X_v := X \times_K K_v\text{,}\) so that \(X_v(K_v) = X(K_v)\text{.}\) In particular, for each \(x \in X(K)\text{,}\) we get a corresponding point \(x_v \in K_v\text{.}\) Meanwhile, for each class \(y \in \Br(X)\text{,}\) and define \(y_v \in \Br(X_v)\) as the image of \(y\) under the functoriality map \(\Br(X_v) \to \Br(X)\text{.}\) We again get a local pairing \(X(K_v) \times \Br(X_v) \to \Br(K_v)\text{.}\) Note that the map \(X(K_v) \to \Br(K_v)\) induced by pairing with a given \(x\) is locally constant for the \(v\)-adic topology on \(X(K_v)\) ([42], Proposition 8.2.9).
Now let \(\inv_v\colon \Br(K_v) \to \QQ/\ZZ\) be the local invariant map at the place \(v\text{.}\) The key point is now to apply Theorem 7.6.12: given a class \(y \in \Br(X)\text{,}\) a necessary condition for a tuple \((x_v)_v \in \prod_v X(K_v)\) to come from a global point \(x \in X(K)\) is that the sum of the local invariants \(\inv_v(x_v, y_n)\) must vanish. That is, each \(y\) gives rise to a commutative diagram as in Figure 7.6.22 in which the vertical maps are evaluation maps and the bottom row is the fundamental exact sequence.
Figure 7.6.22.
Applying this condition for all \(y \in \Br(X)\) picks out a subset of \(\prod_v X(K_v)\text{,}\) the Brauer set, which must contain all of \(X(K)\text{;}\) when this happens, \(X\) is said to have a Brauer-Manin obstruction forcing \(X(K) = \emptyset\text{.}\) More commonly, one says that \(X\) admits a Brauer-Manin obstruction to the local-global principle if each \(X(K_v)\) to be nonempty but the Brauer set is empty. See [42], Chapter 8 for some examples where this occurs.

Exercises Exercises

1.

Let \(K\) be a number field, define the small cyclotomic extension \(K^{\smcy}/K\) as in Definition 7.3.3, let \(S\) be a finite set of finite places of \(K\text{,}\) and let \(m\) be a positive integer. Prove that there exists a subextension \(L\) of \(K^{\smcy}/K\) (which is necessarily cyclic) such that for all \(v \in S\text{,}\) for some place \(w\) of \(L\) above \(K\text{,}\) \([L_w:K_v]\) is divisible by \(m\text{.}\)
Hint.
It suffices to check the case \(K = \QQ\) (with \(m\) replaced by a suitably larger value). We may also assume that \(m\) is a power of some prime \(\ell\text{,}\) and we only need to handle one \(v \in S\text{.}\) For more details, see [37], Lemma VII.7.3.

2.

Let \(D\) be a quaternion algebra over a field \(K\) (see Exercise 5). Prove the following statements directly (without using Lemma 7.6.3).
  1. \(D\) is isomorphic to its opposite algebra.
  2. There is an isomorphism \(D \otimes_K D \cong M_4(K)\) of \(K\)-algebras. Consequently, if \(D\) is not split, then it represents an element of \(\Br(K)\) of order \(2\text{.}\)
Hint.
For the first assertion, check that \(i \mapsto -i, j \mapsto -j, k \mapsto -k\) defines an isomorphism with the opposite algebra.

3.

Let \(K\) be a number field. For \(D\) a quaternion algebra over \(K\text{,}\) let \(S_D\) denote the set of places \(v\) of \(K\) such that \(D \otimes_K K_v\) is split.
  1. Prove that \(S_D\) is finite.
  2. Prove that \(\#S_D \geq 2\text{.}\)
  3. Prove that every two-element set of places of \(v\) occurs as \(S_D\) for some \(D\text{.}\)
Hint.
Deduce everything from Theorem 7.6.12.
PrevTopNext