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).
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