40 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
1.4.3.5. Example. If G = (Gn) is a p-divisible group over S (with height h) and
H = (Hn) is a p-divisible subgroup of G (with height h h) in the sense that
Hn is a closed S-subgroup of Gn compatibly in n, then G/H := (Gn/Hn) is also
a p-divisible group over S. Indeed, a computation with fppf abelian sheaves shows
that the complex
0 G1/H1 Gn+1/Hn+1
[p]
Gn/Hn
is left exact in the sense of fppf abelian sheaves and hence the induced map
(Gn+1/Hn+1)/(G1/H1) Gn/Hn
between finite locally free commutative S-groups is a closed immersion (as is any
proper monomorphism), so it is an isomorphism for order reasons. This shows that
the map [p] : Gn+1/Hn+1 Gn/Hn is faithfully flat with kernel G1/H1 of order
ph−h
, so induction on n implies that
(Gn/Hn)[pm]
is faithfully flat of order
pm(h−h )
for any m n. In particular, the closed immersion Gn/Hn
(Gn+1/Hn+1)[pn]
is
an isomorphism for order reasons, so (Gn/Hn) is a p-divisible group.
In the preceding example, clearly the natural map q : G G/H has functorial
kernel H and has the mapping property of a quotient: any homomorphism of p-
divisible groups G G that kills H uniquely factors through q. Hence, it is
appropriate to define a short exact sequence of p-divisible groups to be a complex
0 G G G 0
such that G is a p-divisible subgroup of G and the induced map G/G G is an
isomorphism, or equivalently the induced complex of finite locally free commutative
S-groups
0 Gn Gn Gn 0
is short exact for all n 1.
For example, if 0 A A A 0 is a short exact sequence abelian
schemes (in the sense of fppf abelian sheaves on the category of S-schemes), or
equivalently A A is faithfully flat with kernel A , or equivalently it is short
exact on geometric fibers over every point of S, then a computation with the snake
lemma for fppf abelian sheaves shows that the induced complex
0 A
[p∞]

A[p∞]
A
[p∞]
0
is short exact. Also, by the definition of the Serre dual p-divisible group in terms of
Cartier duality at finite levels, the Serre dual of a short exact sequence of p-divisible
groups is short exact.
1.4.3.6. Example. An important example of a short exact sequence of p-divisible
groups is the connected-´ etale sequence for a p-divisible group over a complete local
noetherian ring R with residue characteristic p 0. To define this, first recall that
for any finite flat commutative R-group scheme H, the connected component H0 of
the identity section is an open and closed R-subgroup (in particular, it inherits R-
flatness from H, so it is a finite flat R-group) and the associated finite flat quotient
et := H/H0 is finite ´ etale; these properties can be seen via the special fiber. The
short exact sequence
0
H0
H
et
0
of R-group schemes is the connected-´ etale sequence for H.
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