34 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

p-divisible groups, and [41, Ch. II–III] for (contravariant) Dieudonn´ e theory with

applications to p-divisible groups.

1.4.1. Exactness. We shall frequently use exact sequence arguments with abelian

varieties and finite group schemes over fields, as well as with their relative analogues

over more general base schemes. It is assumed that the reader has some familiarity

with these notions, but we now provide a review of this material.

1.4.1.1. Definition. Let S be a scheme, and let T be a Grothendieck topology

on the category of S-schemes. (For our purposes, only the ´ etale, fppf, and fpqc

topologies will arise.) A diagram 1 → G → G

f

→ G → 1 of S-group schemes is

short exact for the topology T if G → G is an isomorphism onto ker(f) and the

map f is a T -covering.

By [30, Exp. IV, 5.1.7.1], in such cases G represents the quotient sheaf G/G

for the chosen Grothendieck topology. By [31, Exp. V, Thm. 4.1(iv), Rem. 5.1], if

G is a quasi-projective group scheme over a noetherian ring R and if G is a finite

flat R-subgroup of G then the fppf quotient sheaf G/G is represented by a quasi-

projective R-group (also denoted G/G ), and the resulting map of group schemes

G → G/G is an fppf G -torsor (so G/G is R-flat if G is).

1.4.1.2. The Cartier dual N

D

of a commutative finite locally free group scheme

N over a base scheme S is the commutative finite locally free group scheme which

represents the fppf sheaf functor Hom(N, Gm) : S HomS -gp(NS , Gm) on the

category of S-schemes. The structure sheaf OND of N D is canonically isomorphic

to the OS-linear dual of the structure sheaf ON of N, and the co-multiplication

(respectively multiplication) map for OND is the OS-linear dual of the multiplication

(respectively co-multiplication) map for ON .

The functor N N

D

on the category of commutative finite locally free group

schemes over S swaps closed immersions and quotient maps, preserves exactness,

and is an involution in the sense that there is a natural isomorphism fN : N

(N

D)D

satisfying (fN

)D

= fND

−1

. See [87, Prop. 2.9] for further details.

As an application, if the S-homomorphism j : G → G is a closed immersion

between finite locally free commutative group schemes then we can use Cartier

duality to give a direct proof that the the fppf quotient sheaf G/G is represented by

a finite locally free S-group (without needing to appeal to general existence results

for quotients by G -actions on quasi-projective S-schemes). The key point is that

the Cartier dual map

jD

:

GD

→ G

D

between finite flat S-schemes is faithfully flat,

as this holds on fibers over S (since injective maps between Hopf algebras over a

field are always faithfully flat [126, 14.1]). Such flatness implies that H :=

ker(jD)

is a finite locally free commutative S-group, so

HD

makes sense and the dual map

q : G

(GD)D

→

HD

is faithfully flat. It is clear that G ⊂ ker(q), and this

inclusion between finite locally free S-schemes is an isomorphism by comparison of

fibral degrees, so

HD

represents G/G .

The following result is useful for constructing commutative group schemes G →

S that are finite and fppf (equivalently, finite and locally free over S).