I am looking for reference or hints how to prove the following result.

Let $G$ be a commutative $S$-group scheme which is the extension of an abelian scheme $A$ by a torus $T$. Then the n-torsion $G[n]$ is a finite flat $S$-group scheme.

Specifically, I have difficulties in showing that $G[n]$ is finite. For a general semi-abelian scheme we know that it is quasi-finite and flat, but not necessarily finite (see e.g. the book Neron Models, Lemma 7.3/2).

Thanks in advance,

  • $\begingroup$ Is $S$ an arbitrary scheme? $\endgroup$
    – Angelo
    Nov 17, 2012 at 20:36
  • $\begingroup$ The statement should be true for an arbitrary scheme but I would be happy with an answer for $S= Spec R$ when $R$ is the ring of integers of a finite $\mathbb{Q}_p$-extension. $\endgroup$ Nov 17, 2012 at 20:48
  • $\begingroup$ I have seen a generalization of this statement in several papers on 1-motives, however no proof or reference is given there. For example: Deligne [Hodge III, 10.1.10] or M. Raynaud, [1-Motifs et Monodromie Géométrique, 3.1]. This is why I think that it should be true. $\endgroup$ Nov 17, 2012 at 20:56
  • $\begingroup$ @Tzanko Matev: In "Néron models", semi-abelian means flat and semi-abelian fibers, the scheme itself is not necessarily an extension of an abelian scheme by a torus. For example if we consider the Néron model of an elliptic curve with multiplicative reduction, it is semi-abelian in the sense of "Néron models", and the $n$-torsion is not finite in general. $\endgroup$
    – Qing Liu
    Nov 17, 2012 at 21:36
  • $\begingroup$ @Qing Liu: I am sorry if the question was not stated well. I know that for a general semi-abelian scheme what I ask is false. I am only interested in the case when the scheme is an extension of an abelian scheme by a torus. $\endgroup$ Nov 18, 2012 at 8:50

1 Answer 1


It is an exercise with descent theory and the snake lemma for fppf abelian group sheaves to deduce the result for $G[n]$ from the cases of $T[n]$ and $A[n]$.

In more detail, by the snake lemma $G[n]$ is an extension of $A[n]$ by $T[n]$ in the sense of such abelian sheaves. Since $A[n]$ and $T[n]$ are each finite fppf over $S$, the same then holds for $G[n]$. Indeed, rather generally, if $$1 \rightarrow G' \rightarrow G \rightarrow G'' \rightarrow 1$$ is a complex of $S$-group schemes with $G'$ affine fppf over $S$ and the diagram is short exact for the fppf topology (so $G'$ is the scheme-theoretic kernel of $G \rightarrow G''$) then the functor of points of $G$ as a $G''$-scheme is a $G'$-torsor for the fppf topology on $G''$, so the $G''$-scheme $G \rightarrow G''$ becomes isomorphic fppf-locally on $G''$ to $G'$ (over the base) as a scheme. Hence, by fppf descent for properties of morphisms, $G \rightarrow G''$ inherits many "nice" properties that may be satisfied by $G' \rightarrow S$, such as: proper, flat, smooth, etale, finite, etc. In particular, $G$ is fppf over $G''$ and if $G'$ is finite over $S$ then so is $G \rightarrow G''$ (and hence so is $G$ if $G''$ is also finite over $S$).

See Oort's LNM book on commutative group schemes for generalizations with the fpqc topology (around section 18, IIRC).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.