1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 55

over κ[ ] and

0 →

Lie(X0)∨ t

→ Lie(E(X0)) → Lie(X0) → 0

over κ.

Grothendieck–Messing theory identifies tX0 with the set of κ[ ]-subbundles of

Lie(E(X0)) ⊗κ κ[ ] which lift

Lie(X0)∨ t

modulo . Such subbundles are parameter-

ized by

Hom(Lie(X0)∨, t

Lie(E(X0)))/

End(Lie(X0)∨) t

=

Hom(Lie(X0)∨, t

Lie(X0))

= Lie(X0)

t

⊗κ Lie(X0)

by assigning to any representative κ-linear map L : Lie(X0)∨ t → Lie(E(X0)) the

κ[ ]-subbundle that is the image of the map Lie(X0)∨ t ⊗κ κ[ ] → Lie(E(X0)) ⊗κ κ[ ]

defined by x + y → x + (y + L(x)) . Our computation of tX0 respects the κ-linear

structure on tX0 , so this completes the proof of Theorem 1.4.4.7.

As an extension of Theorem 1.4.4.7, in Theorem 1.4.5.5 we will establish a

“p-divisible group” version of the second part of Theorem 1.4.4.5. We finish the

present discussion of deformation theory by addressing the useful topic of how

deformation rings behave with respect to “change of coeﬃcients”. This is relevant

when trying to reduce certain structural questions about deformation rings (e.g., is

there a characteristic-0 point?) to the case of an algebraically closed residue field.

1.4.4.12. Example. Consider a local map Λ → Λ between complete local noe-

therian rings with respective residue fields κ and κ , and let X be a proper κ-scheme

such that any proper flat deformations of X over artin local Λ-algebras with residue

field κ admit no non-trivial automorphisms, and similarly for Xκ using Λ -algebras.

The deformation functor DefΛ(X) assigns to any artin local Λ-algebra R with

residue field κ the set of isomorphism classes of proper flat deformations of X over

R, and we define DefΛ (Xκ ) similarly.

It is a theorem of Schlessinger that DefΛ(X) is pro-represented by a complete

local noetherian Λ-algebra R with residue field κ, and likewise DefΛ (Xκ ) is pro-

represented by a complete local noetherian Λ -algebra R with residue field κ . Note

that Λ ⊗ΛR is a complete local noetherian Λ -algebra with residue field κ . By the

local flatness criterion [73, 22.4], base change along R → Λ ⊗ΛR carries flat formal

R-schemes of finite type to flat formal Λ ⊗ΛR-schemes of finite type.

Let X and X be the universal formal deformations of X and Xκ over R and

R respectively. Clearly

X ×Spf(R) Spf(Λ ⊗ΛR) → Spf(Λ ⊗ΛR)

is a proper flat deformation of Xκ , so it arises as the pullback of X along a unique

local Λ -algebra map

(1.4.4.2) R → Λ ⊗ΛR.

Is this map an isomorphism?

To demonstrate the usefulness of an aﬃrmative answer, suppose κ and κ are

perfect of characteristic p 0, and Λ = W (κ) and Λ = W (κ ). (The case κ = κ

will be of most interest.) Consider the problem of whether or not X admits a

proper flat formal lift over a complete local noetherian domain with residue field

κ and characteristic 0. Suppose we can establish that Xκ admits a proper flat

formal lift over a complete local noetherian domain D with residue field κ and

characteristic 0. This lift corresponds to a local W (κ )-algebra homomorphism