1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 35

1.4.1.3. Proposition. Let S be a scheme, and let G and G be finitely presented

separated S-group schemes with G aﬃne and flat over S. For any exact sequence

1 → G → G → G → 1

of group sheaves for the fppf topology on the category of S-schemes, G is represented

by a finitely presented S-group that is flat and aﬃne over G . Moreover, G and

G are S-finite if and only if G is S-finite.

See [87, 17.4] for a generalization (using the fpqc topology).

Proof. For any G -scheme T viewed as an S-scheme, let g ∈ G (T ) correspond

to the given S-morphism T → G . Consider the set Eg (T ) that is the preimage

under G(T ) → G (T ) of g . This is a sheaf of sets on the category of G -schemes

equipped with the fppf topology, and as such it is a left G -torsor (strictly speaking,

a left torsor for the G -group GG ) due to the given exact sequence. In particular,

the fppf sheaves of sets Eg and GG over G are isomorphic fppf-locally over G .

Since G is fppf aﬃne over S and fppf descent is effective for aﬃne morphisms, it

follows that Eg as an fppf sheaf of sets over G is represented by an aﬃne fppf G -

scheme (which is therefore aﬃne fppf over S when G is). It is elementary to check

that this aﬃne G -scheme viewed as an S-scheme has its functor of points naturally

identified with G (since for any S-scheme T and g ∈ G(T ), visibly g ∈ Eg (T ) for

the point g ∈ G (T ) arising from g), so G is represented by an S-group.

Separatedness of G over S and exactness imply that G is closed in G. More-

over, G → G is a left GG -torsor for the fppf topology over G , so it is finite when

G is S-finite. Thus, if G and G are S-finite then G is S-finite. Conversely, if G

is S-finite then its closed subscheme G is S-finite, so the quotient G/G exists as

an S-finite scheme. But G represents this quotient, so G is S-finite too.

1.4.1.4. Remark. If 1 → G → G → G → 1 is an exact sequence of separated

fppf S-groups with G and G abelian schemes then G is an abelian scheme. Indeed,

since G → G is an fppf torsor for the G -group GG that is smooth and proper with

geometrically connected fibers, G → G is smooth and proper with geometrically

connected fibers. The map G → S is also smooth and proper with geometrically

connected fibers, so G → S is as well. Hence, G is an abelian scheme.

It is also true that if G is an abelian scheme and G is a closed S-subgroup of G

that is also an abelian scheme then the fppf quotient sheaf G/G is represented by

an abelian scheme. We will give an elementary proof of this over fields in Lemma

1.7.4.4 using the Poincar´ e reducibility theorem (which is only available over fields).

In general the proof requires a detour through algebraic spaces.

1.4.2. Duality for abelian schemes. In [83, §6.1], duality is developed for pro-

jective abelian schemes, building on the case of an algebraically closed ground

field. Projectivity is imposed primarily due to the projectivity hypotheses in

Grothendieck’s work on Hilbert schemes. The projective case is suﬃcient for our

needs because any abelian scheme over a discrete valuation ring is projective (this

follows from Lemma 2.1.1.1, to which the interested reader may now turn). For

both technical and aesthetic reasons, it is convenient to avoid the projectivity hy-

pothesis. We now sketch Grothendieck’s results on duality in the projective case,

as well as Artin’s improvements that eliminated the projectivity assumption.