1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 43

the Galois action on -adic Tate modules, though the action by F and V is highly

non-trivial even when k =

¯

k (whereas the Galois action on Tate modules is trivial

for such k). As an analogue of the Poincar´ e reducibility theorem for abelian vari-

eties, the isogeny category of p-divisible groups over k is semisimple when k = k;

see Theorem 3.1.3. (This semisimplicity fails more generally, even over finite fields,

as we see in the ´ etale case: Galois groups can have non-semisimple representations

on finite-dimensional Qp-vector spaces.)

To illustrate the use of Dieudonn´ e modules as a replacement for Tate modules

in the case = p, we have the following result that will be important in our later

study of a notion of “complex multiplication” for p-divisible groups.

1.4.3.9. Proposition. Let G be a p-divisible group of height h 0 over a field κ

of characteristic p, and let k be a perfect extension of κ.

(1) If F is a commutative semisimple Qp-subalgebra of

End0(G)

:= Qp⊗Zp End(G)

then [F : Qp] h, with equality if and only if

M∗(Gk)[1/p]

is free of rank 1 as

a W (k) ⊗Zp F -module.

(2) When equality holds, F is its own centralizer in

End0(G).

If moreover the

maximal Zp-order OF in F lies in End(G) then

M∗(Gk)

is free of rank 1 as

a W (k) ⊗Zp OF -module.

In 3.1.8 we will show that End(G) is finitely generated as a Zp-module, but

this fact is not needed here.

Proof. We may and do replace κ with k, so κ is perfect. (In particular, we

may use the notion of isogeny as in 1.4.3.8.) Letting K0 = W (κ)[1/p], we view

M∗(G)[1/p]

as a finite module over the semisimple ring K0 ⊗Qp F . The second

condition in (2) is immediate from the freeness in (1) (as W (κ) ⊗Zp OF is a finite

product of discrete valuation rings that are W (κ)-finite, and

M∗(G)

is finite free

as a W (κ)-module), so it is harmless to pass to an F -linearly isogenous p-divisible

group. Thus, we may decompose G according to the idempotents of F to reduce to

the case when F is a p-adic field. Let k0 be its finite residue field, F0 = W (k0)[1/p],

κ a compositum of k0 with κ over Fp, and K0 = W (κ )[1/p]. Consider the decom-

position

K0 ⊗Qp F = (K0 ⊗Qp F0) ⊗F0 F

j:k0→κ

(K0 ⊗j,F0 F )

where j varies through the embeddings over k0 ∩κ (⊂ k0). This is a finite product of

copies of totally ramified finite extensions of K0, and the factor fields are permuted

transitively by the natural F -linear action of the Galois group Gal(k0/(k0 ∩ κ)).

Note that this Galois group is generated by a power of the absolute Frobenius.

We conclude that any K0 ⊗Qp F -module M canonically decomposes in a com-

patible F -linear way as Mj for vector spaces Mj over the factors fields. Hence,

if M is equipped with an injective F -linear endomorphism F that is semilinear

over the absolute Frobenius of K0 then F must be an F -linear automorphism that

transitively permutes the Mj’s via F -linear isomorphisms. In particular, if such an

M is non-zero then each Mj is a non-zero vector space over the factor field indexed

by j, so M as a K0 ⊗Qp F -module would be free of some positive rank ρ and hence

of K0-dimension [F : Qp]ρ.