42 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

category of left Dk-modules that are finite and free as W -modules; the W -rank of

M∗(G)

is equal to the height of G.

The notion of isogeny for p-divisible groups over a general scheme will be dis-

cussed in 3.3.3–3.3.5, but for our present purposes we only need the case when the

base is a perfect field. This special case is easier to develop, and it is also convenient

to have it available (on geometric fibers) when considering the relative case. Thus,

we now define and briefly study this concept over perfect fields.

1.4.3.8. Definition. A homomorphism f : X → Y between p-divisible groups

over a perfect field K is an isogeny if kerf is a finite K-group and the heights of X

and Y coincide.

We first explain what this means in more concrete terms when char(K) =

p by using p-adic Tate modules, and then the interesting case of perfect K of

characteristic p will proceed similarly by using Dieudonn´ e modules.

Assume char(K) = p, so all p-divisible groups over K are ´ etale (in the sense

that each

pn-torsion

subgroup is ´ etale over K). Via the formation of p-adic Tate

modules, the category of p-divisible groups over K is equivalent to the category of

continuous linear representations of Gal(Ks/K) on finite free Zp-modules. It follows

that f is an isogeny if and only if the induced map between p-adic Tate modules

becomes an isomorphism after inverting p. Thus, a homomorphism f : X → Y

between p-divisible groups is an isogeny if and only if there is a homomorphism

f : Y → X such that f ◦f =

[pn]X

and f ◦f =

[pn]Y

for some n 0, and such an

f is a quotient modulo the finite kernel ker(f) in the sense that any homomorphism

of p-divisible groups X → X over K that kills ker(f) factors uniquely through f.

By forming a quotient Tate module, we likewise see that for any finite K-

subgroup G ⊂ X there is an isogeny of p-divisible groups f : X → Y over K with

ker(f) = G, and f is unique up to unique isomorphism in an evident sense. We call

Y the quotient of X modulo G and denote it as X/G. (It is not entirely trivial to

describe Y [pn] in terms of X and G, and this will make the analogous construction

over a general base scheme less straightforward.)

Now assume char(K) = p (and K is perfect). By arguing with Dieudonn´e

modules in the role of p-adic Tate modules, it is elementary to check that a homo-

morphism f : X → Y between p-divisible groups over K is an isogeny if and only

if the induced map M∗(f) between Dieudonn´ e modules becomes an isomorphism

after inverting p. Consequently, we again obtain that f is an isogeny if and only if

there is a homomorphism f : Y → X such that f ◦f = [pn]X and f ◦f = [pn]Y for

some n 0, and that f has the expected universal mapping property for homomor-

phisms from X that kill the finite kernel of f. Likewise, for any finite K-subgroup

G ⊂ X the induced map of (contravariant!) Dieudonn´ e modules

M∗(X)

→

M∗(G)

is surjective (since G ⊂

X[pn]

for large n), so the kernel of this surjection is W -finite

free of the same rank as

M∗(X).

The corresponding p-divisible group is denoted

X/G because the evident map X → X/G is an isogeny with kernel G.

As with abelian varieties, the isogeny category of p-divisible groups over a per-

fect field k of characteristic p 0 is defined either by formally inverting isogenies or

more concretely by using as the Hom-sets

Hom0(X,

Y ) = Hom(X, Y )[1/p]. For an

abelian variety A of dimension g 0 over k, the Dk-module

M∗(A[p∞])

is finite free

of rank 2g over W , so it is an analogue of the -adic Tate module for = char(k)

even though it is contravariant in A. The Dk-module structure is the analogue of