84 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

between finite K-groups has trivial kernel, so it is a closed immersion. We denote

this closed K-subgroup as

[ϕ](Ar),

so (as with abelian varieties) the map [ϕ] is

faithfully flat onto a closed K-subgroup

[ϕ](Ar)

⊂

As.

Using Lemma 1.7.4.4 in the abelian variety case and the more elementary theory

of quotients for finite commutative K-group schemes in the finite case, the quotient

M ⊗O A := As/[ϕ](Ar) as an abelian variety or finite K-group scheme represents

the cokernel of [ϕ] in the sense of fppf abelian sheaves on the category of K-schemes.

It follows (via the right-exactness of algebraic tensor products) that the K-group

scheme M ⊗O A represents the fppf sheafification of T M ⊗O A(T ).

Now assume that A is an abelian variety. Let M → N be an injective map

between torsion-free O-modules with finite cokernel. There is a map N → M

such that both composites M → M and N → N are multiplication by a common

non-zero integer n. Hence, we get maps in both directions between M ⊗O A and

N ⊗O A whose composites are each equal to multiplication by n, so both maps

between M ⊗O A and N ⊗O A are isogenies.

The assertions concerning O follow by considering the functor T O ⊗O A(T )

and the abelian variety over K representing it.

As an application of the Serre tensor construction with non-projective modules

when the base is a field, we prove a precise form of the “lifting” part of the Deuring

Lifting Theorem:

1.7.4.6. Theorem (Deuring). Let E0 be an elliptic curve over Fq. For any f0 ∈

End(E0) generating an imaginary quadratic field L ⊂

End0(E0)

and p-adic place p

of OL, let R be the valuation ring of the compositum W (Fq)[1/p] · Lp over Qp.

There exists a CM elliptic curve E over R equipped with an endomorphism f

such that (E, f) has special fiber isomorphic to (E0,f0).

By 1.6.2.5, for any E0 over Fq there is an imaginary quadratic field L inside

End0(E0).

The CM structure forces R to have residue field Fq, as we shall see in

the proof below.

Proof. If E0 is ordinary then Lp = Qp and we can choose E to be the Serre–

Tate canonical lift over W (Fq), to which all endomorphisms of E0 uniquely lift;

see 1.4.5.4. Suppose instead that E0 is supersingular. It suﬃces to show that for

any imaginary quadratic field L ⊂

End0(E0)

and O := End(E0) OL, we can lift

(E0,α0) over R where α0 : O → End(E0) is the natural inclusion.

Consider the canonical O-linear isogeny

h : E0 → E0 := OL ⊗O E0

(see 1.7.4.5). The key point is to show that p does not divide the degree of h (so

h induces an isomorphism on p-divisible groups). The degree of h is the order of

the finite Fq-group ker(h), so if this kernel is ´ etale then its order must be relatively

prime to p because a supersingular elliptic curve has infinitesimal p-torsion. Thus,

we may assume ker(h) is not ´ etale, so the infinitesimal identity component of ker(h)

is nontrivial and therefore the relative Frobenius morphism of ker(h) has nontrivial

kernel. The O-linear relative Frobenius morphism FrE0/Fq : E0 →

E0p) (

for the

elliptic curve E0 has kernel of order p, so this latter kernel must lie inside ker(h).