1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 59

There is a pair (G, α) consisting of a p-divisible group G over R and an iso-

morphism α : GK X over K. Moreover, there is a unique isomorphism GR G

recovering the given identification of generic fibers X ⊗K K G ⊗R K .

A special case of this corollary is that any “unramified twist” of the p-adic

Tate module of the generic fiber of a p-divisible group over R is also the p-adic Tate

module of the generic fiber of a p-divisible group over R (since both p-adic Galois

lattices have the same inertial restriction).

Proof. The discrete valuation ring R is the completion of the strict henselization

Rsh

of R inside K, so

Rsh

→ R is a local inclusion with relative ramification degree

1 and induces an isomorphism on residue fields. Thus, for any aﬃne R -scheme Y ,

descent of YK to

Ksh

=

Frac(Rsh)

is equivalent to descent of Y to

Rsh

[10, 6.2,

Prop. D.4]. Applying this to each G

[pn],

we see that G uniquely descends to a

p-divisible group G over

Rsh

compatibly with the descent XKsh of GK .

For each σ ∈

Gal(Ksh/K)

and the associated continuous K-automorphism

σ of the completion K , the canonical isomorphism

σ∗(XKsh

) XKsh induces

an isomorphism σ

∗

(GK ) GK . By Tate’s full faithfulness result (applied over

R ), this extends to an isomorphism σ

∗

(G ) G of p-divisible groups over R .

The uniqueness of the descent from R to

Rsh

implies that this latter isomorphism

uniquely descends to an

Rsh-isomorphism

ασ :

σ∗(G

) G extending the canonical

isomorphism

σ∗(XKsh

) XKsh .

The 1-cocycle condition ατ ◦ τ

∗(ασ)

= ατσ over

Rsh

is inherited from the

generic fiber. Thus, since

Rsh

is a direct limit of R-subalgebras that are Galois

local finite ´ etale over R, on each finite

pn-torsion

level the

Rsh-isomorphisms

ασ

amount to an ´ etale descent datum relative to R → Rsh [10, 6.2/B]. The resulting

effective descent of G to a p-divisible group G over R is equipped with a canonical

K-isomorphism α : GK X. By Tate’s full faithfulness theorem applied over R ,

the K -isomorphism

(GR ) ⊗R K = GK ⊗K K XK G ⊗R K

uniquely extends to an R -isomorphism GR G .

The link between the deformation theories of abelian varieties and p-divisible

groups in characteristic p is provided by the Serre-Tate deformation theorem:

1.4.5.3. Theorem (Serre–Tate). Let R be a ring in which a prime p is nilpotent,

and let I be an ideal in R such that

In

= 0 for some n 1. Define R0 = R/I, and

for an abelian scheme A and p-divisible group G over R let A0 and G0 denote their

respective reductions modulo I.

For any abelian scheme A over R, let

A

: A[p∞]0 A0[p∞] denote the canon-

ical isomorphism. The functor A (A0,A[p∞],

A

) from the category of abelian

schemes over R to the category of triples (A0,G, : A[p∞]0 G0) is an equivalence.

See [57, 1.2.1] for a proof of this result.

The most important application of the Serre-Tate deformation theorem is that

for an abelian variety A0 over a field of characteristic p 0, the infinitesimal

deformation theory of A0 coincides with that of its p-divisible group. Likewise,

if we fix a subring O ⊂ End(A0) or a polarization of A0 (or both) then via the

injection End(A0) ⊂

End(A0[p∞])

(Proposition 1.2.5.1) and the identification of