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 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
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