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
Previous Page Next Page