From: George McNinch
Subject: Re: Algebraic Groups
Date: 28 Dec 1999 16:45:42 -0500
Newsgroups: sci.math.research
Keywords: algebraic groups resemble linear groups and finite groups
>>>>> "Victor" == Victor S Miller writes:
Victor> Can someone point me to a proof of the theorem that a
Victor> group variety (or probably a group scheme, maybe subject
Victor> to some reasonable hypotheses to rule out pathological
Victor> cases) can be formed by extensions of finite groups,
Victor> abelian varieties and matrix groups (something which
Victor> becomes isomorphic to the group of n by n matrices after
Victor> possibly a finite extension of the defining field)?
If the group variety is affine, such a group is an extension of a
connected linear group by a finite group. Here, a linear group just
means a closed subgroup of GL_n (n by n non-singular matrices).
This amounts to the statement that the connected component of the
identity has finite index in G (since G is quasicompact), and that an
affine group variety has a faithful matrix representation. For these
facts, you can see e.g. A. Borel, Linear Algebraic Groups,
Springer-Verlag GTM 126.
But the connected part needn't become isomorphic to some GL_n after a
field extension, or even be built up by extensions of such groups
(e.g. the "additive group" will not, nor will various classical groups
like the symplectic group Sp_2n, ...)
One can say more about the connected component group, though. Namely,
it has a maximal normal, connected, solvable subgroup for which the
quotient group is more-or-less a product of various (almost) simple
groups (like Sp_2n, SL_n, SO_n, ...) For this, see again Borel's book.
When G is no longer affine, I think that J.-P. Serre, "Algebraic
Groups and Class Fields" may contain the statement you want (I could
be mistaken as I don't have it handy right now). But I think at least
Serre should have there a reference for what you want.
--
George McNinch | __O
www.nd.edu/~gmcninch | _-\<,_
Dept. Math, Univ Notre Dame | (_)/ (_)
|--------------
==============================================================================
From: David Joyner
Subject: Re: Algebraic Groups
Date: Tue, 28 Dec 1999 15:24:58 -0500
Newsgroups: sci.math.research
"Victor S. Miller" wrote:
[original post quoted --djr]
Chevalley's Theorem, Ch III, section 3.3, in Shaferavich, B.A.G., 1977
==============================================================================
From: Axel Schmitz-Tewes
Subject: Re: Algebraic Groups
Date: Wed, 29 Dec 1999 10:01:50 +0100
Newsgroups: sci.math.research
George McNinch schrieb:
> ....
>
> When G is no longer affine, I think that J.-P. Serre, "Algebraic
> Groups and Class Fields" may contain the statement you want (I could
> be mistaken as I don't have it handy right now).
If you have a group variety G, you can look at the co-group \Gamma of
global sections of the structure sheaf. From that you get a closed
immersion
Spec(\Gamma) -> G
of group varieties which imbeds Spec(\Gamma) as a normal subgroup. The
quotient (for the existence there is something to prove!) should be
an abelian variety and Spec(\Gamma) is an affine variety for which you
can
use the results posted before.
regards,
axel