1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 49

dim(X0)

t

= h − d by [119, §2.3, Prop. 3]). A deformation of X0 over a complete

local noetherian ring R with residue field κ is a pair (X , i) consisting of a p-divisible

group X over R and an isomorphism i : Xκ X0.

There is an evident notion of morphism between such pairs. By Proposition

1.4.4.3, deformations of X0 have no non-trivial automorphisms (lifting the identity

on X0). Hence, it is reasonable to study the functor DefΛ(X0) assigning to any

R ∈ CΛ the set of isomorphism classes of deformations of X0 over R. In contrast

with the deformation theory of non-zero abelian varieties, for which the universal

formal deformation is never algebraizable beyond the case of elliptic curves, the

universal formal deformation of a p-divisible group is also a universal deformation

relative to all complete local noetherian Λ-algebras with residue field κ since p-

divisible groups are built from torsion-levels that are finite flat over the base.

1.4.4.7. Theorem. The functor DefΛ(X0) is pro-represented by a power series

ring over Λ in d(h−d) variables. The tangent space tX0 to this functor is canonically

isomorphic to Lie(X0)

t

⊗κ Lie(X0) as a κ-vector space.

The pro-representability for connected X0 over perfect κ is established in [124]

by using formal group laws to verify Schlessinger’s criteria; the perfectness is re-

quired to carry out a Dieudonn´ e module computation establishing that dimκ(tX0 )

∞ (equal to d(h − d)). This approach does not prove formal smoothness. Over al-

gebraically closed fields the pro-representability for general X0 is deduced formally

from the connected case in the proof of [16, Thm. 4.4 (2)]. The general case over

any κ may be deduced from Schlessinger’s criteria and [51, 4.4]; the latter ingredi-

ent is proved via the cotangent complex (also see [51, 4.8] for perfect κ). For the

convenience of the reader, here is a proof for general κ that avoids the machinery

of the cotangent complex.

Proof. First, we address the formal smoothness. The case of connected X0 is

a special case of the unobstructedness of lifting commutative formal Lie groups,

which can be proved over any ring via Cartier theory; see [136, Thm. 4.46]. For

disconnected X0 consider a deformation X of X0 over an artinian local Λ-algebra

R with residue field κ. There is a unique (up to unique isomorphism) ´ etale p-

divisible group E over R that lifts X0

´ et

over κ, so

X´ et

is uniquely isomorphic to E

as deformations of X0

´ et

.

Since

X0 is a deformation of the identity component of X0, we see that the

construction of such X comes in two steps: (i) deform X0 to a (necessarily con-

nected) p-divisible group over R (this step is unobstructed, by the settled connected

case), and (ii) construct extensions over R of E by the chosen deformation of X0 0.

Such extensions in the sense of fppf abelian sheaves on the category CR of finite

R-algebras (equipped with the fppf topology) arise from p-divisible groups, due to:

1.4.4.8. Lemma. Let R be an local artinian ring with residue characteristic p 0,

and choose a connected p-divisible group G over R and an ´ etale p-divisible group E

over R. For any short exact sequence of fppf abelian sheaves

(1.4.4.1) 0 → G → Y → E → 0

on the category CR of finite R-algebras (with the fppf topology), Y is a p-divisible

group (so the given short exact sequence is the connected-´ etale sequence of Y ).