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 [Δ : 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: