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.
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