1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 39

functor on finite commutative group schemes. In particular,

M∗(G(p))

σ∗(M∗(G))

as W -modules.

(3) Let FrG/k : G →

G(p)

be the relative Frobenius morphism. The σ-semilinear

action on

M∗(G)

induced by

M∗(FrG/k)

through the isomorphism

M∗(G(p))

σ∗(M∗(G))

equals the action of F, and G is connected if and only if F is

nilpotent on

M∗(G).

(4) There is a natural k-linear isomorphism M∗(G)/FM∗(G) Lie(G)∨ respect-

ing extension of the perfect base field.

(5) For the Cartier dual GD, naturally M∗(GD) HomW (M∗(G),K/W ) with

K = W [1/p], using the operators F( ) : m → σ( (V(m))) and V( ) : m →

σ−1( (F(m))) on K/W -valued linear forms .

For an abelian scheme A → S with fibers of constant dimension g 1 and

its finite commutative pn-torsion subgroup scheme A[pn] with order (pn)2g, the

directed system

A[p∞]

:=

(A[pn])n

1

satisfies the following definition (with h = 2g).

1.4.3.3. Definition. A p-divisible group of height h 0 over a scheme S is a

directed system G = (Gn)n

1

of commutative S-groups Gn such that: Gn is killed

by

pn,

each Gn → S is finite and locally free, [p]: Gn+1 → Gn is faithfully flat for

every n 1, G1 → S has constant degree

ph,

and Gn is identified with

Gn+1[pn]

for all n 1.

The (Serre) dual p-divisible group

Gt

is the directed system (Gn

D)

of Cartier

dual group schemes Gn

D

with the transition maps Gn

D

→ Gn+1

D

that are Cartier

dual to the quotient maps [p] : Gn+1 → Gn.

As an illustration, if A → S is an abelian scheme with fibers of dimension

g 1 then the isomorphisms

A[n]D At[n]

respecting multiplicative change in

n (as noted immediately below Theorem 1.4.2.5) yield a canonical isomorphism

between the Serre dual

A[p∞]t and the p-divisible group At[p∞] of the dual abelian

scheme At (see [86, Prop. 1.8] or [87, Thm. 19.1]).

1.4.3.4. Remark. In view of the sign discrepancy for comparisons of double du-

ality in Theorem 1.4.2.5, if ϕ : A → At is an S-homomorphism and

f :

A[p∞]

→

At[p∞] A[p∞]t

is the associated homomorphism between p-divisible groups then the dual homo-

morphism ϕt : A → At (strictly speaking, ϕt ◦ ιA/S via double duality for abelian

schemes) has as its associated homomorphism

A[p∞]

→

A[p∞]t

the

negative1

of f

t

(using double duality for p-divisible groups).

It follows that if ϕ is symmetric with respect to double duality for abelian

schemes then f is skew-symmetric with respect to double duality for p-divisible

groups. The converse is also true: we can see immediately via skew-symmetry

of f that ϕ and

ϕt

induce the same homomorphism between p-divisible groups,

and to conclude that ϕ =

ϕt it suﬃces to check on fibers due to the rigidity of

abelian schemes (as in 1.4.2.2). On fibers we can apply the faithfulness of passage

to p-divisible groups over fields via 1.2.5.1 with = p.

1A

related sign issue in the double duality for commutative finite group schemes over perfect

fields is discussed in a footnote at the end of B.3.5.5.