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))