1.4.
DIEUDONN´
E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 57
of the tangent space of the deformation functor of a proper κ-scheme Y0 with
H1(Y0, (ΩY0/κ)∨). 1
In the case of p-divisible groups it follows from the compatibility
of Grothendieck–Messing theory with respect to base change.
Deformation rings for geometric objects equipped with extra structure (e.g.,
polarized abelian varieties with endomorphism structure) are not as easy to describe
as in the formally smooth setting used in the preceding proof. In such cases we can
sometimes use a global moduli scheme to establish an isomorphism result for the
“change of coefficients” map:
1.4.4.14. Proposition. Let Λ be a complete local noetherian ring with residue
field κ, and let (A0,φ0,α0) be a polarized abelian variety of dimension g 0 with
endomorphism structure α0 : O End(A0) for a Z-finite associative ring O. The
analogue of (1.4.4.2) for (A0,φ0,α0) is an isomorphism for any Λ .
The same “change of coefficients” isomorphism holds for the formal deforma-
tion ring of any pair (A0,α0).
Proof. First we treat the polarized cases by relating the deformation problem
to global moduli schemes via auxiliary “level structure”, and then we modify the
method to apply in the absence of polarizations. The introduction of auxiliary level
structure can be carried out without increasing κ by using non-constant “finite ´etale
level structure” arising inside A0, as follows.
Fix an integer n 3 not divisible by char(κ). The finite ´ etale group scheme
A0[n] over κ uniquely lifts to a finite ´ etale group scheme G over Λ. Consider the
functor F on the category of Λ-schemes that assigns to any Λ-scheme S the set of
isomorphism classes of quadruples (A, φ, α, τ) where (A, φ) is a polarized abelian
scheme over S of relative dimension g with φ of constant square degree deg(φ0),
α : O End(A) is a homomorphism, and τ : A[n] GS is an S-group isomorphism
(“level structure”). These quadruples have no nontrivial automorphisms since n 3
and n Λ×.
By standard moduli space arguments (using Hilbert schemes), the functor F is
represented by a Λ-scheme M locally of finite type. The triple (A0,φ0,α0) and the
canonical isomorphism τ0 : A0[n] define a point ξ M(κ), and the formal
deformation ring for (A0,φ0,α0) is naturally isomorphic to the completed local
ring OM,ξ

at ξ (since the finite ´ etale “level structure” τ0 uniquely lifts through any
infinitesimal deformation).
Since represents the restriction of F to the category of Λ -schemes, it is
straightforward to verify that the analogue of (1.4.4.2) for the present situation is
the inverse of the natural isomorphism
Λ ⊗ΛOM,ξ

OMΛ

,ξκ
,
so it is an isomorphism. (This style of argument applies whenever we can relate
the infinitesimal deformation problem to the formal structure on a global moduli
scheme over Λ.)
Now we treat the case of formal deformation rings for (A0,α0) and (A0,α0)
relative to some Λ Λ . The absence of a polarization eliminates the option to
use global moduli schemes for abelian schemes. Instead, we work with formal Hom-
schemes attached to formal abelian schemes. (The same procedure can be used to
handle the polarized case above.) Let A Spf(R) be the universal deformation
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