18 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
action of AutF (Matn(F )), so it is useful to determine this automorphism group.
The subgroup of inner automorphisms is GLn(F )/F
×
, arising from conjugation
against elements of
Matn(F = GLn(F ). In general, the inner automorphisms
are the only ones:
1.2.3.2. Theorem (Skolem–Noether). For a central simple algebra D over a field
F , the inclusion D×/F × AutF (D) carrying u to (d udu−1) is an
equality. That is, all automorphisms are inner.
Proof. This is [53, §4.6, Cor. to Thm. 4.9].
We finish our discussion of central simple algebras by using the Skolem–Noether
theorem to build the K-linear reduced trace map TrdD/K : D K for a central
simple algebra D over a field K.
1.2.3.3. Construction. Let D be a central simple algebra over an arbitrary field
K. It splits over a separable closure Ks, which is to say that there is a Ks-algebra
isomorphism f : DKs Matn(Ks) onto the n × n matrix algebra for some n 1.
By the Skolem-Noether theorem, all automorphisms of a matrix algebra are given
by conjugation by an invertible matrix. Hence, f is well-defined up to composition
with an inner automorphism.
The matrix trace map Tr : Matn(Ks) Ks is invariant under inner automor-
phisms and is equivariant for the natural action of Gal(Ks/K), so the composition of
the matrix trace with f is a Ks-linear map DKs Ks that is independent of f and
Gal(Ks/K)-equivariant. Thus, this descends to a K-linear map TrdD/K : D K
that is defined to be the reduced trace. In other words, the reduced trace map is
a twisted form of the usual matrix trace, just as D is a twisted form of a matrix
algebra. (For d D, the K-linear left multiplication map x d · x on D has
trace [D : K]TrdD/K (x), as we can see by scalar extension to Ks and a direct
computation for matrix algebras. The elimination of the coefficient [D : K] is the
reason for the word “reduced”.)
1.2.4. Brauer groups. For applications to abelian varieties it is important to
classify division algebras of finite dimension over Q (such as the endomorphism
algebra of a simple abelian variety over a field). If Δ is such a ring then its center
Z is a number field and Δ is a central division algebra over Z. More generally, the
set of isomorphism classes of central division algebras over an arbitrary field has an
interesting abelian group structure. This comes out of the following definition.
1.2.4.1. Definition. Central simple algebras D and D over a field K are similar
if there exist n, n 1 such that the central simple K-algebras D ⊗K Matn(K) =
Matn(D) and D ⊗K Matn (K) = Matn (D ) are K-isomorphic.
The Brauer group Br(K) is the set of similarity classes of central simple algebras
over K, and [D] denotes the similarity class of D. For classes [D] and [D ], define
[D][D ] := [D ⊗K D ].
This composition law on Br(K) is well-defined and makes it into an abelian
group with inversion given by
[D]−1
=
[Dopp],
where
Dopp
is the “opposite algebra”.
By Proposition 1.2.2.2, each element in Br(K) is represented (up to isomorphism)
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