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