PlanetMath: group variety

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group variety

(Definition)

Let be a variety (an affine, projective, or quasi-projective variety). We say is a group variety if is provided with morphisms of varieties:

$\displaystyle \mu:G\times G$	$\displaystyle \to G$
$\displaystyle (g_1,g_2)$	$\displaystyle \mapsto g_1g_2,$

$\displaystyle \iota: G$	$\displaystyle \to G$
$\displaystyle g$	$\displaystyle \mapsto g^{-1},$

and

$\displaystyle \epsilon: \{*\}$	$\displaystyle \to G$
$\displaystyle *$	$\displaystyle \mapsto e,$

and if these morphisms make the elements of

into a group.

In short, should be a group object in the category of varieties. Examples include the general linear group of dimension on and elliptic curves.

Group varieties that are actually projective are in fact abelian groups (although this is not obvious) and are called abelian varieties; their study is of interest to number theorists (among others).

Just as schemes generalize varieties, group schemes generalize group varieties. When dealing with situations in positive characteristic, or with families of group varieties, often they are more appropriate.

There is also a (not very closely related) concept in group theory of a “variety of groups”.

"group variety" is owned by archibal.

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Keywords:

variety

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Cross-references: characteristic, positive, group schemes, schemes, number, abelian varieties, obvious, abelian groups, elliptic curves, dimension, general linear group, category, group object, group, morphisms, quasi-projective variety, variety
There is 1 reference to this entry.

This is version 1 of group variety, born on 2004-02-16.
Object id is 5582, canonical name is GroupVariety.
Accessed 1740 times total.

Classification:

AMS MSC:	14L10 (Algebraic geometry :: Algebraic groups :: Group varieties)
	14K99 (Algebraic geometry :: Abelian varieties and schemes :: Miscellaneous)
	20G15 (Group theory and generalizations :: Linear algebraic groups :: Linear algebraic groups over arbitrary fields)

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