20 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

1.2.4.3. Theorem. Let L be a global field. There is an exact sequence

0 Br(L)

v

Br(Lv)

∑

invLv

Q/Z 0

where the direct sum is taken over all places of L and the first map is defined via

extension of scalars.

Proof. This is [120, §9.7, §11.2].

For a global field L and central division algebra Δ over L, invv(Δ) denotes

invLv (ΔLv ). Theorem 1.2.4.3 says that a central division algebra Δ over a global

field L is uniquely determined up to isomorphism by its invariants invv(Δ), and

that these may be arbitrarily assigned subject to the conditions invv(Δ) = 0 for all

but finitely many v and

∑

invv(Δ) = 0. Moreover, the order of [Δ] in Br(L) is the

least common “denominator” of the local invariants invv(Δ) ∈ Q/Z.

If K is any field then for a class c ∈ Br(K) its period is its order and its index is

[Δ : K] with Δ the unique central division algebra over K representing the class

c. It is a classical fact that the period divides that index and that these integers

have the same prime factors (see [107, X.5], especially Lemma 1 and Exercise 3),

but in general equality does not hold. For example, there are function fields of

complex 3-folds for which some order-2 elements in the Brauer group cannot be

represented by a quaternion algebra; examples are given in [61, §4], and there are

examples with less interesting fields as first discovered by Brauer. We have noted

above that over local fields there is equality of period and index (the archimedean

case being trivial). The following deep result is an analogue over global fields.

1.2.4.4. Theorem. For a central division algebra Δ over a global field L, the order

of [Δ] in Br(L) is [Δ : L].

As a special (and very important) case, elements of order 2 in Br(L) are pre-

cisely the Brauer classes of quaternion division algebras for a global field L; as noted

above, this fails for more general fields. Since Theorem 1.2.4.4 does not seem to

be explicitly stated in any of the standard modern references on class field theory

(though there is an allusion to it at the end of [4, Ch. X, §2]), and the structure

theory of endomorphism algebras of abelian varieties rests on it, here is a proof.

Proof. Let Δ have degree n2 over L and let d be the order of [Δ] in Br(L), so d|n.

Note that d is the least common multiple of the local orders dv of [ΔLv ] ∈ Br(Lv)

for each place v of L, with dv = 1 for complex v, dv|2 for real v, and dv = 1 for all

but finitely many v. Using these formal properties of the dv’s, we may call upon

the full power of global class field theory via Theorem 6 in [4, Ch. X] to infer the

existence of a cyclic extension L /L of degree d such that [Lv : Lv] is a multiple of

dv for every place v of L (here, v is any place on L over v, and the constraint on the

local degree is only non-trivial when dv 1). In the special case d = 2 (the only

case we will require) one only needs weak approximation and Krasner’s Lemma

rather than class field theory: take L to split a separable quadratic polynomial

over L that closely approximates ones that define quadratic separable extensions of

Lv for each v such that dv = 2.

By (1.2.4.1), restriction maps on local Brauer groups induce multiplication by

the local degree on the local invariants, so ΔL is locally split at all places of L .

Thus, by the injectivity of the map from the global Brauer group into the direct