1.7. A THEOREM OF GROTHENDIECK AND A CONSTRUCTION OF SERRE 83
Proof. Recall from the Poincar´ e reducibility theorem over K (see Theorem 1.2.1.3)
that there is an abelian subvariety B B over K that is an isogeny complement
to B in the sense that the natural map f : B × B B is an isogeny. Since f is a
finite flat surjection, so B (B × B )/ker(f) as fppf abelian sheaves, we likewise
have B/B = B /(B B ) as fppf sheaves. Thus, the problem for B/B is the
same as for the quotient of B by the finite K-subgroup scheme B B . Hence,
it suffices to show that B /G is (represented by) an abelian variety for any finite
K-subgroup G B .
Rather than appealing to existence results for quotients by the free action of a
finite group scheme on a quasi-projective scheme, here is a more direct argument
via fppf descent theory and a special property of abelian varieties: for any n 1,
the map [n]B : B B is a finite flat surjection, so B /B [n] B as fppf
abelian sheaves. To exploit this, note that the K-group G is killed by its order n.
(Indeed, we may assume char(K) = p 0 and use the connected-´ etale sequence for
G and kernels of relative Frobenius morphisms to reduce to the case when FrG/K
vanishes. In such cases p kills G since [p]G = VerG(p)/K ◦FrG/K .) Thus, G B [n],
so as fppf abelian sheaves B /G is a B [n]/G-torsor over B /B [n] B .
Since B [n]/G is represented by a finite K-scheme, by effective descent for
finite morphisms we see that the quotient sheaf B /G is therefore represented by
a finite flat B -scheme over which B is a finite flat cover (even a G-torsor). This
implies that B /G is proper, smooth, and connected, so it is an abelian variety, as
desired.
Here is the promised generalized Serre tensor construction over fields (allowing
non-projective modules).
1.7.4.5. Proposition. Let A be an abelian variety or finite commutative group
scheme over a field K. Let O End(A) be a homomorphism from a commutative
ring. For any finitely generated O-module M, the functor T M ⊗O A(T ) on K-
schemes has fppf sheafification that is respectively represented by an abelian variety
or finite commutative group scheme M ⊗O A.
Suppose A is an abelian variety. For an injective map M N between torsion-
free O-modules with finite cokernel, the induced map M ⊗O A N ⊗O A is an
isogeny. In particular, if O is a Z-flat O-algebra that is finitely generated as an O-
module and OQ OQ is an isomorphism then the natural map of abelian varieties
A A := O ⊗O A is an isogeny and the identification
End0(A)
=
End0(A
)
carries O
End0(A)
into End(A ).
The notation O ⊗O A should not be confused with the standard notation for
affine base change of schemes. Also, if M is killed by a non-zero element of O then
it is killed by a non-zero integer and hence the abelian variety M ⊗O A vanishes.
Proof. Choose a finite presentation of O-modules
Or
ϕ
−→
Os
−→ M 0.
The map ϕ is given by an s × r matrix over O, so it defines an analogous map
[ϕ] :
Ar

As
between K-groups. Since we are working over a field, if A is an
abelian variety then the map [ϕ] has image that is an abelian subvariety of
As
onto
which [ϕ] is faithfully flat. If instead A is a finite commutative K-group then there
is a finite flat quotient map
Ar

Ar/ker[ϕ].
The induced map
Ar/ker[ϕ]

As
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