58 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

of A0, so for R := Λ ⊗ΛR it follows from Proposition 1.4.4.13 that the universal

deformation of A0 is the base change

A = A ×Spf(R) Spf(R ) → Spf(R ).

For any formal abelian schemes B and C over a complete local noetherian ring

R, applying the arguments with Hom-functors from the proof of Theorem 1.4.4.5

over the artinian quotients of R proves that the Hom-functor Hom(B, C) : Y

HomY-gp(BY, CY) on the category of formal (adic) R-schemes Y locally of finite

type is represented by a separated and locally finite type formal R-scheme.

Thus, we get a separated and locally finite type formal R-scheme End(A) clas-

sifying endomorphisms of A, and its formation commutes with local base change to

any complete local noetherian ring (such as R ). The definition of an action on (a

base change of) A by the Z-finite ring O underlying α0 amounts to giving several

points of End(A) satisfying finitely many relations. These relations correspond to a

formal closed subscheme in a fiber power of End(A), so we get an adic formal mod-

uli scheme M locally of finite type over R that classifies O-actions on A. Moreover,

the formation of M commutes with base change along any local homomorphism

from R to another complete local noetherian ring.

The given α0

corresponds to a rational point ξ ∈ M(κ) in the special fiber,

and the deformation ring pro-representing DefΛ(A0,α0) is the completed local ring

OM,ξ. ∧ Similarly, M := MR contains the κ -point ξ in its special fiber that corre-

sponds to α0 and arises by base change from ξ, so there is a natural isomorphism

Λ ⊗ΛOM,ξ

∧

OM

∧

,ξ

.

The inverse of this isomorphism is the “change of coeﬃcients” map that we wanted

to prove is an isomorphism.

1.4.5. Hodge–Tate decomposition and Serre–Tate lifts. We finish our sum-

mary of the theory of p-divisible groups by recording (for later reference) two fun-

damental theorems. The first is a deep result of Tate.

1.4.5.1. Theorem (Tate). Let R be a complete discrete valuation ring with perfect

residue field of characteristic p 0 and fraction field K of characteristic 0. For

any p-divisible groups G and G over R, the natural injective map Hom(G, G ) →

Hom(GK , GK) is bijective. Moreover, if CK denotes the completion of an alge-

braic closure K of K then there is a canonical CK -linear Gal(K/K)-equivariant

isomorphism

(1.4.5.1) CK ⊗Qp Vp(GK) (CK(1) ⊗K Lie(G)K) ⊕ (CK ⊗K

Lie(Gt)K ∨

).

Proof. The full faithfulness of G GK is [119, 4.2], and the construction of

the isomorphism (1.4.5.1) occupies most of [119].

A useful application of the full faithfulness in Tate’s theorem is “completed

unramified descent” for p-divisible groups:

1.4.5.2. Corollary. Let R be as in Theorem 1.4.5.1 and let K be the completion

of the maximal unramified extension of K inside K. Let G be a p-divisible group

over the valuation ring R of K , and assume that the generic fiber GK is equipped

with a descent to a p-divisible group X over K.