82 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

whose second map is injective (Proposition 1.2.5.1) and composition is the canonical

homomorphism that is injective. It therefore suﬃces to show that the composite

map is an isomorphism. But M is free of rank 1 as an OL,-module, so it is

equivalent to show that the natural map

OL, → EndOL, (A[

∞

])

is an isomorphism for all .

The case = char(K) is trivial, as then the -adic Tate module T (A) is free

of rank 1 over OL, (since V (A) is free of rank 1 over L , due to faithfulness and

Q -dimension reasons). Now assume char(K) = p 0 and = p. Decomposing

A[p∞] according to the primitive idempotents of OL,p, for each p-adic place v of L

the v-factor has height at least [Lv : Qp] (by Proposition 1.4.3.9) and hence height

exactly [Lv : Qp]. Thus, it suﬃces to prove more generally that if X is a p-divisible

group over K of height h 0 and F is a p-adic field of degree h over Qp such that

OF ⊂ End(X) then OF is its own centralizer in End(X).

We may and do assume that K is algebraically closed, so the Dieudonn´ e module

M∗(X)

makes sense and is free of rank 1 as a W (K) ⊗Zp OF -module (Proposition

1.4.3.9). Thus, any OF -linear endomorphism f of X gives rise to a W (K) ⊗Zp OF -

linear endomorphism of

M∗(X),

so

M∗(f)

must be multiplication by some c ∈

W (K) ⊗Zp OF . But

M∗(f)

commutes with the F operator on

M∗(X),

so c is

invariant under the absolute Frobenius automorphism σ of W (K). This forces

c ∈ W

(K)σ=1

⊗Zp OF = OF , so f is an OF -multiplier, as desired.

1.7.4.3. Remark. The O-linear projectivity hypothesis on M in the construction

of M ⊗O A cannot be dropped, even when the base is the spectrum of a discrete

valuation ring. For example, if R is a p-adic discrete valuation ring and E is an

elliptic curve over R with endomorphism ring O = Z[p

√

−p] (as can be easily con-

structed using classical CM theory for elliptic curves), then for the non-projective

O-module M = Z[

√

−p] the fppf sheafification of the functor T M ⊗O E(T ) on

R-schemes is not representable. (The idea is as follows. First one proves that a

representing object, if one exists, must be an elliptic curve E . By presenting M

over O using two generators and two relations, we get a quotient homomorphism

E × E → E whose kernel must be an R-flat divisor in E × E. Studying its defining

equation in the formal group of E × E leads to a contradiction.)

For any homomorphism f : A → A between abelian varieties over a field,

the image f(A) is an abelian subvariety over A and the map A → f(A) is flat.

Such good properties for f(A) generally fail for homomorphisms between abelian

schemes over a more general base, but their availability over fields enables us to

push through the initial cokernel idea for the Serre tensor construction over fields.

In this way we can avoid dualizing M and hence make the construction work with

weaker hypotheses on M than projectivity. We shall now give a version of this for

abelian varieties (and in 4.3.1 there is a version for p-divisible groups), but first we

require a general lemma:

1.7.4.4. Lemma. For an abelian variety B over a field K and an abelian subvari-

ety B , the quotient sheaf B/B for the fppf topology on the category of K-schemes

is represented by an abelian variety.