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 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 over R and
R respectively. Clearly
X ×Spf(R) Spf(Λ ⊗ΛR) Spf(Λ ⊗ΛR)
is a proper flat deformation of , 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 ). (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 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 )-algebra homomorphism
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