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.
Previous Page Next Page