|
|
|
|
group scheme
|
(Definition)
|
|
|
A group scheme is a group object in the category of schemes. Similarly, if $S$ is a scheme, a <</SPAN>#49#>group scheme over $S$ is a group object in the category of schemes over $S$
As usual with schemes, the points of a group scheme are not the whole story. For example, a group scheme may have only one point over its field of definition and yet not be trivial. The points of the underlying topological space do not form a group under the obvious choice for a group law.
We can view a group scheme $G$ as a ``group machine'': given a ring $R$ the set of $R$ points of $G$ forms a group. If $S$ is a scheme that is not affine, we can nevertheless interpret $G$ as a family of groups fibred over $S$
|
"group scheme" is owned by archibal.
|
|
(view preamble | get metadata)
Cross-references: ring, obvious, group, topological space, field, points, schemes, category, group object
There are 12 references to this entry.
This is version 1 of group scheme, born on 2004-02-24.
Object id is 5616, canonical name is GroupScheme.
Accessed 4871 times total.
Classification:
| AMS MSC: | 14K99 (Algebraic geometry :: Abelian varieties and schemes :: Miscellaneous) | | | 14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms) | | | 14L10 (Algebraic geometry :: Algebraic groups :: Group varieties) | | | 20G15 (Group theory and generalizations :: Linear algebraic groups :: Linear algebraic groups over arbitrary fields) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
