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

G´ et

:=

(G´ et

n

) is a

p-divisible group (called the ´ etale part of G). We call

0 →

G0

→ G →

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