36 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

1.4.2.1. Let A → S be an abelian scheme, and let PicA/S be the functor assigning

to any S-scheme T the group of isomorphism classes of pairs (L , i) consisting of

an invertible sheaf L on AT and a trivialization i : eT

∗

(L ) OT along the identity

section eT of AT → T . This is an fppf group sheaf on the category of S-schemes,

and its restriction to the category of S -schemes (for an S-scheme S ) is PicAS

/S

.

Let PicA/S

0

⊂ PicA/S be the subfunctor classifying pairs (L , i) that lie in the

identity component of the Picard scheme on geometric fibers. By [7, Exp. XIII,

Thm. 4.7(i)] (see [39, §9.6] for the projective case), the inclusion j : PicA/S

0

→

PicA/S is an open subfunctor; i.e., it is relatively representable by open immersions.

This means that for any S-scheme T and (L , i) ∈ PicA/S(T ), PicA/S

0

×PicA/S T as

a functor on T -schemes is represented by an open subscheme U ⊂ T ; explicitly,

there is an open subscheme U ⊂ T such that a T -scheme T lies over U if and only

if the T -pullback of (L , i) lies in

Pic0

on geometric fibers over T .

By Grothendieck’s work on Picard schemes (see [39, Part 5]), if A → S is

projective Zariski-locally on S then PicA/S is represented by a locally finitely pre-

sented and separated S-scheme and the open subscheme

At

representing

PicA/S0

is quasi-projective Zariski-locally on S and finitely presented. For noetherian S,

functorial criteria show that

At

is proper and smooth (see [83, §6.1]), hence an

abelian scheme; the case of general S (with A projective Zariski-locally on S) then

follows by descent to the noetherian case.

To drop the projectivity hypothesis, one has to use algebraic spaces. Infor-

mally, an algebraic space over S is an fppf sheaf on the category of S-schemes that

is “well-approximated” by a representable functor (relative to the ´ etale topology),

so concepts from algebraic geometry such as smoothness, properness, and connect-

edness can be defined and behave as expected; see [60]. By Artin’s work on relative

Picard functors as algebraic spaces (see [5, Thm. 7.3]), PicA/S is always a separated

algebraic space locally of finite presentation, and by [7, Exp. XIII, Thm. 4.7(iii)]

the open algebraic subspace PicA/S

0

is finitely presented over S.

The functorial arguments that prove smoothness and properness for

PicA/S0

when A is projective work without projectivity because the same criteria are avail-

able for algebraic spaces. Thus, PicA/S

0

is smooth and proper over S in the sense

of algebraic spaces. Consequently, by a theorem of Raynaud (see [38, Thm. 1.9]),

PicA/S

0

is represented by an S-scheme At; this must be an abelian scheme, called

the dual abelian scheme. Its formation commutes with any base change on S, and

it is contravariant in A in an evident manner.

1.4.2.2. Over A ×

At

there is a Poincar´ e bundle PA/S provided by the universal

property of

At,

exactly as in the theory of duality for abelian varieties over a field.

In particular, PA/S is canonically trivialized along e × idAt . Let e ∈

At(S)

be the

identity, so for any S-scheme T the point eT ∈ At(T ) corresponds to OAT equipped

with the canonical trivialization of eT ∗ (OAT ). Thus, setting T = A gives that PA/S

is also canonically trivialized along idA × e . Hence, the pullback of PA/S along

the flip

At

× A A ×

At

defines a canonical S-morphism ιA/S : A →

Att.

This

morphism carries the identity to the identity, so it is a homomorphism. By applying

the duality theory over fields to the fibers of A over S, it follows that ιA/S is an

isomorphism; in other words, the pullback of PA/S along the flip At ×A A×At is

uniquely isomorphic to PAt/S respecting trivializations along the identity sections

of both factors. Such uniqueness implies that ιA/S

t

is inverse to ιAt/S.