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


.
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.
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