Is the n-torsion of an extension of an abelian variety by a torus, finite and flat? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:51:14Z http://mathoverflow.net/feeds/question/112710 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112710/is-the-n-torsion-of-an-extension-of-an-abelian-variety-by-a-torus-finite-and-fla Is the n-torsion of an extension of an abelian variety by a torus, finite and flat? Tzanko Matev 2012-11-17T19:32:31Z 2012-11-18T08:48:30Z <p>I am looking for reference or hints how to prove the following result. </p> <blockquote> <p>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. </p> </blockquote> <p>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).</p> <p>Thanks in advance,</p> http://mathoverflow.net/questions/112710/is-the-n-torsion-of-an-extension-of-an-abelian-variety-by-a-torus-finite-and-fla/112718#112718 Answer by nosr for Is the n-torsion of an extension of an abelian variety by a torus, finite and flat? nosr 2012-11-17T21:18:14Z 2012-11-17T21:18:14Z <p>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 $A[n]$.</p> <p>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 <em>fppf-locally on $G''$</em> 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 $S$).</p> <p>See Oort's LNM book on commutative group schemes for generalizations with the fpqc topology (around section 18, IIRC).</p>