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

H´ et := H/H0 is finite ´ etale; these properties can be seen via the special fiber. The

short exact sequence

0 →

H0

→ H →

H´ et

→ 0

of R-group schemes is the connected-´ etale sequence for H.