46 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

monomials lying in (pI,

I2).

Iterating, if f ∈ Hom(Γ , Γ) vanishes over κ then

f ◦

[pn]Γ

vanishes modulo the ideal Jn, where J0 = m and Jn+1 = (pJn,Jn).

2

We

have already seen that JN = 0 for suﬃciently large N, so f

◦[pN

]Γ = 0 for large N.

By hypothesis the isogenous endomorphism

[pN

]Γ = [p]Γ

N

of Γ induces an injective

endomorphism of the coordinate ring, so f = 0.

An important fact in the study of lifting problems for abelian varieties from

characteristic p to characteristic 0 is that infinitesimal lifting for such an abelian

variety is entirely controlled by that of its p-divisible group. This will be made

precise in Theorem 1.4.5.3 (and Example 1.4.5.4 will address algebraization aspects

in the limit). We now focus on the existence and structure of deformation rings for

abelian varieties and p-divisible groups as well as the behavior of these deformation

rings relative to extension of the residue field.

1.4.4.4. Definition. Let A0 be an abelian variety of dimension g over a field κ.

For a complete local noetherian ring R with residue field κ, a deformation of A0

over R is a pair (A, i) consisting of an abelian scheme A over R and an isomorphism

i : Aκ A0 over κ.

There is an evident notion of isomorphism between two deformations of A0

over R, and such deformations have no non-trivial automorphisms. Likewise, if

A0 is equipped with a polarization φ0 : A0 → A0

t

or an injective homomorphism

α0 : O → End(A0) from a specified Z-finite associative ring O (or both!), we define

in an evident way the notion of deformation for A0 equipped with this additional

structure. In the case of polarizations, any R-homomorphism φ : A →

At

lifting φ0

is necessarily a polarization. Indeed, the symmetry of φ is inherited from φ0 (due

to faithfulness of passage to the special fiber for abelian schemes over a local ring),

and the ampleness on A of the pullback

(1,φ)∗(PA)

of the Poincar´ e bundle PA

is inherited from the ampleness on A0 of its restriction

(1,φ0)∗(PA0

) due to [34,

IV3, 9.6.4].

Fix a complete local noetherian ring Λ with residue field κ (e.g., a Cohen ring

for κ), and let CΛ be the category of artinian local Λ-algebras R with local structure

map Λ → R and residue field κ. The deformation functor

DefΛ(A0) : CΛ → Set

assigns to every R in CΛ the set of isomorphism classes of deformations of A0 over

R. Likewise, if A0 is equipped with a polarization φ0 : A0 → A0 t and endomorphism

structure α0 : O → End(A0) (for a Z-finite associative ring O) then we define the

deformation functor DefΛ(A0,φ0,α0) similarly. This is a subfunctor of DefΛ(A0).

A covariant functor F : CΛ → Set is pro-representable if there is a complete

local noetherian Λ-algebra R with local structure map Λ → R and residue field

κ such that F HomΛ(R, ·) (using local Λ-algebra homomorphisms). A formal

deformation ring for A0 (if one exists) is an R that pro-represents DefΛ(A0). The

reason we say “formal” is that over such an R there is merely a universal formal

abelian scheme (which is however universal modulo mR

n

among abelian scheme

deformations of A0 over objects in CΛ whose maximal ideal has vanishing nth

power, for each n 1).

When we include a polarization as part of the deformation problem, if this

enhanced problem admits a pro-representing ring R then by Grothendieck’s formal