80 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
a “tensor product” against a finite-index extension of coefficient rings. There is
a construction of this sort due to Serre [104], applicable over any base scheme,
though it turns out to not be applicable to the above situation because OP is not a
projective O-module when O = OP . We wish to adapt Serre’s construction to the
above situation over a field, so we digress to explain Serre’s procedure.
1.7.4. Serre’s tensor construction. Consider a scheme S, a commutative ring
O, and an O-module scheme A over S. (The main example of such an A to keep
in mind is an abelian scheme, but there are other interesting examples, such the
n-torsion subgroups of such abelian schemes for n 1.) Let M be a projective
O-module of finite rank.
The projectivity of M ensures that the functor T M ⊗O A(T ) on S-schemes
is represented by an S-scheme, denoted M ⊗O A, and that M ⊗O A inherits many
nice properties from A such as flatness, smoothness, properness, good behavior
with respect to analytification over C, etc. The interested reader can see [22, §7]
for details (where non-commutative O are also considered), but the idea of the
construction of M ⊗O A is simple, as follows.
If
Or
ϕ

Os
M 0 is a presentation then we want to define M ⊗O A
to be the cokernel of the S-group map
Ar

As
induced by the matrix of ϕ.
Without the projectivity assumption on the O-module M, over a general base
scheme S such a quotient may not exist. However, since M is locally free of finite
rank as an O-module (by the projectivity hypothesis) we can instead begin with a
presentation of the dual module M

and then dualize to get a left-exact sequence
0 M
Os

Or
with suitable local splitting properties to enable us to construct
M ⊗O A as a scheme-theoretic kernel. More explicitly, by projectivity there is an
integer r 1 and O-module M such that
Or
M M , so there is an O-
linear idempotent endomorphism e of
Or
such that M = ker(e). The kernel of the
associated O-linear endomorphism of Ar represents M ⊗O A.
1.7.4.1. Example. Let L be a CM field, and let (A, i) and (A , i ) be CM abelian
varieties over a field K, where i and i respectively define complex multiplication
by L. Assume that via these embeddings, OL lies in the endomorphism rings of
the abelian varieties. Finally, assume that there exists a non-zero OL-linear map
A A. (By Proposition 1.5.4.1, when K is algebraically closed of characteristic 0
it is equivalent to assume that the associated CM types Φ and Φ on L coincide.)
We claim that M := Hom((A , i ), (A, i)) is an invertible OL-module whose
formation is unaffected by extension of the ground field and that if char(K) = 0
then the evaluation map
M ⊗OL A A
is an isomorphism. (Hence, over an algebraically closed field of characteristic 0
the Serre tensor construction defines a natural transitive action of the finite group
Pic(OL) on the set of isomorphism classes of CM abelian varieties with a fixed CM
type (L, Φ) and CM order OL. The argument below will show that the action is
simply transitive.)
The non-zero L-vector space MQ =
Hom0((A
, i ), (A, i)) has dimension exactly
1 since for = char(K) the natural map
Q ⊗Q MQ HomL (V (A ),V (A))
Previous Page Next Page