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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,

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  • $\begingroup$ Is $S$ an arbitrary scheme? $\endgroup$
    – Angelo
    Commented 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. $\endgroup$
    – Tzanko Matev
    Commented 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. $\endgroup$
    – Tzanko Matev
    Commented 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
    Commented 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. $\endgroup$
    – Tzanko Matev
    Commented Nov 18, 2012 at 8:50

1 Answer 1

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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).

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