58 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
of A0, so for R := Λ ⊗ΛR it follows from Proposition 126.96.36.199 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 188.8.131.52
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
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.
184.108.40.206. 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
(220.127.116.11) CK ⊗Qp Vp(GK) (CK(1) ⊗K Lie(G)K) ⊕ (CK ⊗K
Proof. The full faithfulness of G GK is [119, 4.2], and the construction of
the isomorphism (18.104.22.168) occupies most of .
A useful application of the full faithfulness in Tate’s theorem is “completed
unramified descent” for p-divisible groups:
22.214.171.124. Corollary. Let R be as in Theorem 126.96.36.199 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.