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
of R inside K, so
→ 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
is equivalent to descent of Y to
Prop. D.4]. Applying this to each G
we see that G uniquely descends to a
p-divisible group G over
compatibly with the descent XKsh of GK .
For each σ ∈
and the associated continuous K-automorphism
σ of the completion K , the canonical isomorphism
) 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
implies that this latter isomorphism
uniquely descends to an
) G extending the canonical
) XKsh .
The 1-cocycle condition ατ ◦ τ
= ατσ over
is inherited from the
generic fiber. Thus, since
is a direct limit of R-subalgebras that are Galois
local finite ´ etale over R, on each finite
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:
184.108.40.206. 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
= 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[p∞]0 A0[p∞] denote the canon-
ical isomorphism. The functor A (A0,A[p∞],
) 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) ⊂
(Proposition 220.127.116.11) and the identification of