Talk:Group scheme

Untitled

Regarding the fact that matrix inversion is polynomial, wouldn't the usual Cayley-Hamilton argument make this clearer? Michael Kinyon 16:49, 11 April 2006 (UTC)[reply]

Well, OK, I guess since Cayley-Hamilton follows from Cramer's Rule, it's not really all that different. Still, there is something about the way the argument is written here that is unsatisfactory. I just can't quite put my finger on it. Michael Kinyon 16:53, 11 April 2006 (UTC)[reply]

Another construction (perhaps a more natural one) interprets GL(n) as a group in 2n^2 coordinates, namely as a group of mutually inverse pairs of n by n matrices. In other words: GL(n)={(A,B)|AB=1}. Matrix inversion is just swapping coordinates and hence obviously polynomial. Lenthe 09:48, 30 November 2006 (UTC)[reply]

Commutative and abelian

Some changes have been made in the article. The terminology is tricky because abelian scheme is a concept derived from abelian variety, and therefore it is usual to say "commutative group scheme". Charles Matthews (talk) 16:55, 14 September 2009 (UTC)[reply]

I am not too sure, but-

In the definition, instead of "presheaf corresponding to G under the Yoneda embedding", should it be the "S-scheme corresponding to G under the Yoneda embedding"? —Preceding unsigned comment added by 92.45.136.158 (talk) 00:21, 12 November 2009 (UTC)[reply]

No, it looks right as it is. As defined, G is an S-scheme, the presheaf is the image of G under the Yoneda embedding, i.e. G considered as a representable functor from (S-scheme)op to Sets. It is just saying that the Yoneda embedding factors through a functor from (S-scheme)op to Groups, that the S-scheme morphisms into the S-scheme G functorially form a group. More concretely, just saying that G(R), the R-valued points of G, the solutions of the systems of equations defining G in a ring R (it is enough to take (affine schemes)op = rings as where we are evaluating G) nicely form groups with arrows going where they should when we have ring homomorphisms, e.g. G(Z)-->G(Q)-->G(R)-->G(C) or G(Z) --> G(Z/pZ). The book of Demazure and Gabriel I added to the refs is a very good place to learn such things.John Z (talk) 02:03, 12 November 2009 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Group scheme/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Field now geometry. Could be also Algebra, but classified this way in anticipation of a separate Algebraic Geometry field. Stca74 06:59, 31 May 2007 (UTC)[reply]

Last edited at 06:59, 31 May 2007 (UTC). Substituted at 02:10, 5 May 2016 (UTC)

Formatting needs to be fixed

The formatting on this page should be upgraded to latex.

Abelian Varieties Needed!

This page would benefit from explicit models of higher dimensional abelian varieties *not* constructed from products of elliptic curves.