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
= 0 for large N.
By hypothesis the isogenous endomorphism
[pN
= [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 : 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 be the category of artinian local Λ-algebras R with local structure
map Λ R and residue field κ. The deformation functor
DefΛ(A0) : Set
assigns to every R in 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 : 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 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
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