1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 61

is ordinary when

A0[p∞]

has the maximal possible p-rank, namely g; equivalently,

A0[p∞]0

has height g.

Now assume that A0 is ordinary. Since

A0[p∞] t A0[p∞]t

=

(A0[p∞]0)t

×

(A0[p∞]´ et )t

with A0

t

isogenous to A0, for height reasons it follows that the dual of

A0[p∞]0

is

´ etale. In other words, if A0 is ordinary then canonically A0[p∞]

X0t

× X0 for

´ etale p-divisible groups X0 and X0 that are functorial in A0[p∞]. For any complete

local noetherian ring R with residue field k, X0 and X0 uniquely lift to respective

´ etale p-divisible groups X and X over R, so the deformation X

t

× X of

X0t

× X0

corresponds to a canonical formal deformation A of A0 over R. We claim that the

formal abelian scheme A is algebraic; its algebraization is called the Serre–Tate

canonical lifting.

Choose a polarization φ0 : A0 → A0.

t

The skew-symmetry of the associated

quasi-polarization

X0t

× X0 =

A0[p∞]

→

A0[p∞] t A0[p∞]t

= X0

t

× X0,

forces it to have the form −f0

t

× f0 for a homomorphism f0 : X0 → X0. There

is a unique lifting f : X → X of f0 since X0 and X0 are ´ etale, so −f

t

× f lifts

−f0

t

× f0. In other words, the map induced by φ0 between p-divisible groups lifts

(necessarily uniquely, by 1.4.4.3) to a homomorphism

A[p∞]

→

At[p∞],

as suﬃces

for the algebraicity of A.

Quasi-polarizations yield a p-divisible group analogue of Proposition 1.4.4.14:

1.4.5.5. Theorem. Let Λ be a complete local noetherian ring with residue field

κ of characteristic p 0, let X0 be a p-divisible group over κ, and let α0 : O →

End(X0) be an injective homomorphism from an associative finite flat Zp-algebra.

Let φ0 : X0 → X0 t be a quasi-polarization of X0.

(1) The functors DefΛ(X0,α0) and DefΛ(X0,φ0,α0) on CΛ are pro-represented by

quotients of the deformation ring for DefΛ(X0).

(2) Let Λ → Λ be a local map between complete local noetherian rings, with κ → κ

the induced map between residue fields. Let (X0,φ0,α0) = (X0,φ0,α0)κ , and

let R and R be the respective rings pro-representing DefΛ(X0,φ0,α0) and

DefΛ (X0,φ0,α0). The natural map

(1.4.5.2) R → Λ ⊗ΛR

analogous to (1.4.4.2) is an isomorphism. The same holds for the deformation

rings of (X0,α0) and (X0,α0), as well as for (X0,φ0) and (X0,φ0).

Proof. Since α0 is encoded in terms of finitely many endomorphisms of X0, and

φ0 is a homomorphism X0 → X0,

t

for the proof of (1) it suﬃces to establish the

following general claim (applied to the universal deformation of X0 and its dual).

Let X and Y be p-divisible groups over a complete local noetherian ring (R, m)

with residue field κ of characteristic p, and let f0 : X0 → Y0 be a homomorphism

between the special fibers. We claim there exists an ideal I ⊂ R such that for any

local map R → R to an artinian local ring with residue field κ, a lift XR → YR of

f0 exists if and only if I has vanishing image in R.