1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 37

A homomorphism f : A →

At

is symmetric when the map

f

t

◦ ιA/S : A

Att

→

At

is equal to f. Writing f

†

:= f

t◦ιA/S,

the equality ιA/S

t

= ιAt/S

−1

and the functoriality

of ιA/S in A (applied with respect to f) implies f †† = f, so if we abuse notation by

writing f t rather than f † then (f t)t = f. We say f is symmetric when f t = f (or

more accurately, f † = f). This property holds if it does so on fibers over S, because

homomorphisms f, f : A ⇒ B between abelian schemes coincide if fs = fs for all

s ∈ S. Indeed, for noetherian S such rigidity is [83, Cor. 6.2], and the general case

reduces to this because equality on all fibers descends through direct limits (since

it says that the finitely presented ideal of (f, f )−1(ΔA/S) in OA is nilpotent).

A polarization of A is a homomorphism f : A →

At

that is a polarization on

geometric fibers. Any such f is necessarily symmetric. The properties of polar-

izations are developed in [83, §6.2] for projective abelian schemes, but the only

purpose of imposing projectivity at the outset (even though it is a consequence of

the definition, due to [34, IV3, 9.6.4]) is to ensure the existence of the dual abelian

scheme, so such an assumption may be eliminated.

1.4.2.3. Definition. A homomorphism ϕ : A → B between abelian schemes over

a scheme S is an isogeny when it is surjective with finite fibers. (Equivalently, the

homomorphims ϕs are isogenies in the sense of abelian varieties for each s ∈ S.)

Since quasi-finite proper morphisms are finite by [34, IV4, 18.12.4] (or by [34,

IV3, 8.11.1] with finite presentation hypothesis, which suﬃces for us), any isogeny

between abelian schemes is a finite morphism. Moreover, by the fibral flatness

criterion [34, IV3, 11.3.11], such maps are flat. Hence, if ϕ as above is an isogeny

then it is finite locally free (and surjective), so the closed subgroup ker(ϕ) is a finite

locally free commutative S-group scheme. Thus, B represents the fppf quotient

sheaf A/ker(ϕ). For example, setting ϕ = [n]A for n 1 gives A/A[n] A.

Turning this around, suppose we are given the abelian scheme A and a closed

S-subgroup N ⊂ A that is finite locally free over S. Consider the fppf quotient

sheaf A/N. We claim that this quotient is (represented by) an abelian scheme, so

the map A → A/N with kernel N is an isogeny. It suﬃces to work Zariski-locally

on S, so we may assume that N → S has all fibers with the same order n 1. We

then have N ⊂ A[n], due to the following result (proved in [123, §1]):

1.4.2.4. Theorem (Deligne). Let S be a scheme and let H be a commutative S-

group scheme for which the structural morphism H → S is finite and locally free.

If the fibers Hs

have rank n for all s ∈ S then H is killed by n.

The quotient sheaf A/N is an fppf torsor over A/A[n] A with fppf covering group

A[n]/N that is finite (and hence aﬃne) over S. It then follows from effective fppf

descent for aﬃne morphisms that the quotient A/N is represented by a scheme

finite over A/A[n] = A, and the map A → A/N is an fppf A[n]/N-torsor, so the

S-proper S-smooth A is finite locally free over A/N (as A[n]/N is finite locally free

over S). Hence, A/N is proper and smooth since A is, and likewise its fibers over

S are geometrically connected. Thus, A/N is an abelian scheme as desired.