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 : 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 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
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 ).
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