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 : Gκ 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 MΛ 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