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 )[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.
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