1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 21
sum of the local ones (for L ) we conclude that the Galois extension L /L of degree
d splits Δ. (The existence of cyclic splitting fields for all Brauer classes is proved
for number fields in [120] and is proved for all global fields in [128], but neither
reference seems to control the degree of the global cyclic extension.) It is a general
fact for Brauer groups of arbitrary fields [107, Ch. X, §5, Lemma 1] that every
Brauer class split by a Galois extension of degree r is represented by a central
simple algebra with degree r2. Applying this fact from algebra in our situation,
[Δ] = [D] for a central simple algebra D of degree d2 over L. But each Brauer class
is represented by a unique central division algebra, and so D must be L-isomorphic
to a matrix algebra over Δ. Since [D : L] = d2 and : L] = n2 with d|n, this
forces d = n as desired.
1.2.5. Homomorphisms and isotypicity. The study of maps between abelian
varieties over a field rests on the following useful injectivity result.
1.2.5.1. Proposition. Let A and B be abelian varieties over a field K. For any
prime (allowing = char(K)), the natural map
Z ⊗Z Hom(A, B) Hom(A[
∞],B[ ∞])
is injective, where the target is the Z -module of maps of -divisible groups over K
(i.e., compatible systems of K-group maps A[
n]
B[
n]
for all n 1).
Proof. Without loss of generality, K is algebraically closed (and hence perfect).
When = char(K) the assertion is a reformulation of the well-known analogous
injectivity with -adic Tate modules (and such injectivity in turn underlies the
proof of Z-module finiteness of Hom(A, B)). The proof in terms of Tate modules
is given in [82, §19, Thm. 3] for = char(K), and when phrased in terms of -
divisible groups it works even when = p = char(K) 0. For the convenience
of the reader, we now provide the argument for = p in such terms. We will use
that the torsion-free Z-module Hom(A, B) is finitely generated, and our argument
works for any (especially = char(K)).
Choose a Z-basis {f1,...,fn} of Hom(A, B). For c1,...,cn Z it suffices to
show that if

cifi kills A[ ] then |ci for all i. Indeed, if we can prove this then
consider the case when

cifi kills A[
∞].
Certainly ci = ci for some ci Z ,
and (

cifi) · kills A[ n] for all n 0. But the map A[ n] A[ n−1] induced by
-multiplication is faithfully flat since it is the pullback along A[ n−1] A of the
faithfully flat map : A A, so

cifi kills A[ n−1] for all n 0. In other words,
the kernel of the map in the Proposition would be -divisible, yet this kernel is a
finitely generated Z -module, so it would vanish as desired.
Now consider c1,...,cn Z such that

cifi kills A[ ]. For the purpose of
proving ci Z for all i, it is harmless to add to each ci any element of Z . Hence,
we may and do assume ci Z for all i, so

cifi : A B makes sense and kills
A[ ]. Since : A A is a faithfully flat homomorphism with kernel A[ ], by fppf
descent theory any K-group scheme homomorphism A G that kills A[ ] factors
through : A A (see [30, IV, 5.1.7.1] and [98]). Thus,∑

cifi = · h for some
h Hom(A, B). Writing h =

mifi with mi Z, we get ci ⊗fi = ·

1⊗mifi
in Z ⊗Z Hom(A, B). This implies ci = mi for all i, so we are done.
A weakening of simplicity that is sometimes convenient is:
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