1.4.
DIEUDONN´
E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 41
A p-divisible group G = (Gn) over R is connected if each Gn is connected;
equivalently, every Gn has infinitesimal special fiber. By a snake lemma argument
with fppf abelian sheaves and the connected-´ etale sequence for finite flat commu-
tative R-group schemes, if G = (Gn) is a p-divisible group over R then
G0
:=
(Gn)0
is a p-divisible group (called the connected component of G) and
et
:=
(G´ et
n
) is a
p-divisible group (called the ´ etale part of G). We call
0
G0
G
et
0
the connected-´ etale sequence for G.
Somewhat deeper lies the fact (see [119, §2.2] and [75, II, 3.3.18, 4.5]) that
if G is a connected p-divisible group over R then O(G) := lim

O(Gn) is a formal
power series ring in finitely many variables over R such that the induced formal
R-group structure makes
[p]∗
: O(G) O(G) finite flat, and moreover there is an
equivalence
G G := Spf(O(G))
from the category of connected p-divisible groups over R to the category of com-
mutative formal Lie groups Γ over R for which [p]Γ is an isogeny. The quasi-inverse
functor is Γ (Γ[pn]). This equivalence defines the (relative) dimension and Lie
algebra for a connected p-divisible group over R, via analogous notions for formal
Lie groups over R. By [119, §2.3, Prop. 3], dim(G) +
dim(Gt)
is the height of G.
For later purposes, here is how this construction works in the important exam-
ple of the p-divisible group G =
A[p∞]
arising from an abelian R-scheme A. What
is G? Let CR denote the category of artinian local R-algebras that are module-finite
over R (and hence killed by some power of the maximal ideal of R). Every point
of A valued in such an algebra and supported at the identity of the special fiber is
a point of the commutative formal Lie group A := Spf(OA,0).

A computation with
formal group laws shows that all such points have p-power torsion, due to R having
residue characteristic p. (This calculation will be given in a self-contained manner
in the proof of Proposition 1.4.4.3.) Thus, for any C CR, the formal Lie group
A has each of its C-points supported in some
A[pn]0.
It follows that A and G pro-
represent the same functor on CR, so the natural map G A is an isomorphism.
In particular, the p-divisible group of A has the same (relative) dimension and Lie
algebra as A does.
1.4.3.7. Now consider p-divisible groups over a perfect field k of characteristic
p 0. For any p-divisible group G = (Gn)n
1
over k with height h 1 we
let
M∗(G)
denote the Dk-module lim

M∗(Gn).
By the same style of arguments
used to work out the Z -module structure of Tate modules of abelian varieties
in characteristic = (resting on knowledge of the orders of the -power torsion
subgroups), we use W -length to replace counting to infer that
M∗(G)
is a free
W -module of rank h and
M∗(G)/pnM∗(G)

M∗(Gn)
is an isomorphism for all n 1. The p-divisible group G is connected if and only
if F is topologically nilpotent on
M∗(G)
(since this is equivalent to the nilpotence
of F on each
M∗(Gn)).
The Dieudonn´ e module functor defines an anti-equivalence between the cate-
gory of p-divisible groups over k (using the evident notion of morphism) and the
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