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).
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