38 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

1.4.2.5. Theorem. Let ϕ : A → B be an isogeny between abelian schemes over a

scheme S, and let N = ker(ϕ). Duality applied to the exact sequence

0 → N −→ A

ϕ

−→ B → 0

functorially yields an exact sequence

0 → N

D

−→

Bt

ϕt

−→

At

→ 0.

That is, the map

ϕt

is an isogeny whose kernel is canonically isomorphic to N

D.

Moreover, double duality for abelian schemes and for finite locally free commu-

tative group schemes are compatible up to a sign: if we identify ϕ and

ϕtt

via ιA/S

and ιB/S then the natural isomorphism (N

D)D ker((ϕt)t)

=

ker(ϕtt)

ker(ϕ) =

N is the negative of the canonical isomorphism provided by Cartier duality.

We refer the reader to [86, Thm. 1.1, Cor. 1.3] for a proof based on arguments

that relativize the ones over an algebraically closed field in [82]. (An alternative

approach, at least for the first part, is [87, Thm. 19.1], resting on the link between

dual abelian schemes and Ext-sheaves given in [87, Thm. 18.1].) The special case

ϕ = [n]A : A → A implies that naturally

A[n]D

=

At[n]

for every n 1 because

[n]A

t

= [n]At (by [87, 18.3]); this identification respects multiplicative change in n.

1.4.3. Constructions and definitions. Let us now focus on constructions spe-

cific to the theory of finite commutative group schemes over a perfect field k of

characteristic p 0. Let W = W (k) be the ring of Witt vectors of k; e.g., if k is

finite of size q =

pr

then W is the ring of integers in an unramified extension of

Qp of degree r. Let σ be the unique automorphism of W that reduces to the map

x →

xp

on the residue field k.

1.4.3.1. Definition. The Dieudonn´ e ring Dk over k is W [F, V], where F and V

are indeterminates subject to the relations

(1) FV = VF = p,

(2) Fc = σ(c)F and cV = Vσ(c) for all c ∈ W .

Explicitly, elements of Dk have unique expressions as finite sums

a0 +

j0

ajFj

+

j0

bj

Vj

with coeﬃcients in W (so the center of Dk is clearly

Zp[Fr, Vr]

if k has finite size

pr

and it is Zp otherwise; i.e., if k is infinite).

Some of the main conclusions in classical Dieudonn´ e theory, as developed from

scratch in [41, Ch. I–III], are summarized in the following theorem.

1.4.3.2. Theorem. There is an additive anti-equivalence of categories G

M∗(G)

from the category of finite commutative k-group schemes of p-power order to the

category of left Dk-modules of finite W -length. Moreover, the following hold.

(1) A group scheme G has order p

W

(M∗(G)),

where

W

(·) denotes W -length.

(2) If k → k is an extension of perfect fields with associated extension W → W

of Witt rings (e.g., the absolute Frobenius automorphism of k) then the functor

W ⊗W (·) on Dieudonn´ e modules is naturally identified with the base-change