1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 47

GAGA algebraization theorems [34, III1, 5.4.1, 5.4.5] there is a universal defor-

mation (i.e., a polarized abelian scheme deformation over R that represents the

deformation functor on the category of all complete local noetherian Λ-algebras

with residue field κ).

1.4.4.5. Theorem. The deformation functor

DefΛ(A0) is pro-representable and

formally smooth, with tangent space canonically isomorphic to Lie(A0) t ⊗κ Lie(A0)

as a κ-vector space. In particular, this formal deformation ring is a formal power

series ring over Λ in g2 variables.

Each deformation functor DefΛ(A0,φ0,α0) and DefΛ(A0,α0) is pro-represented

by a quotient of the formal deformation ring for A0.

Proof. The case of DefΛ(A0) is due to Grothendieck and is explained in detail

in [88, Thm. 2.2.1] (where the description of the tangent space to the deforma-

tion functor is given in the proof). To show that a deformation functor F of the

form DefΛ(A0,φ0,α0) or DefΛ(A0,α0) is pro-represented by a quotient of the for-

mal deformation ring (R, m) for A0, for each integer n 1 we consider the full

subcategory CΛ,n of objects R ∈ CΛ whose maximal ideal has vanishing nth power.

The restriction DefΛ(A0)|CΛ,n is represented by

R/mn.

For each n, suppose there

is an ideal In ⊂

R/mn

such that

(R/mn)/In

represents the subfunctor F |CΛ,n of

DefΛ(A0)|CΛ,n . Since CΛ,n is a full subcategory of CΛ,n+1, by universality we see

that In+1 has image In under R/mn+1 → R/mn. Hence, there is a unique ideal

I ⊂ R such that In = (I + mn)/mn for all n 1, so R/I is the desired quotient.

We are reduced to the following general problem for abelian schemes (applied

to the universal deformation of A0 over R/mn for every n 1 and the structure

(φ0,α0) on its reduction A0 modulo the nilpotent ideal m/mn). Let A and B

be abelian schemes over a noetherian scheme S, and let I ⊂ OS be a nilpotent

coherent ideal sheaf defining a closed subscheme S0 ⊂ S. For a homomorphism

f0 : A0 → B0 over S0, the condition on an S-scheme T that (f0)T0 lifts (necessarily

uniquely!) to a T -homomorphism AT → BT is represented by a closed subscheme

of S (visibly containing S0). We will prove this by using Hom-schemes.

Consider the functor Hom(A, B) : T HomT -gp(AT , BT ) on S-schemes. We

shall prove this is represented by an S-scheme locally of finite type (avoiding projec-

tivity hypotheses on A and B). Grothendieck’s construction of Hom-schemes from

Hilbert schemes (via graph arguments) for schemes that are proper, flat, and finitely

presented over the base requires projectivity because this hypothesis is needed to

ensure representability of Hilbert functors. But Artin showed (see [5, Cor. 6.2])

that the Hilbert functor of a proper, flat, and finitely presented S-scheme X is an

algebraic space that is separated and locally of finite type over S. Consequently, the

same holds for Hom-functors between such schemes, and so also for the subfunctors

that impose compatibility with group laws.

We conclude that H :=

Hom(A, B) is an algebraic space that is separated and

locally of finite type over S. For all s ∈ S the fibers Hs are ´ etale (by the functorial

criterion), and an algebraic space that is separated and locally quasi-finite over a

noetherian scheme is a scheme [60, II, 6.16]. Thus, H is represented by a separated

and locally finite type S-scheme that we denote also as H.

The given f0 defines a section to H0 := H×S S0 → S0. We claim that the closed

subscheme Z0 → H0 underlying this section is stable under generization. Suppose

not, so there exists a discrete valuation ring R and an element h0 ∈ H0(R) whose