group scheme

A group scheme is a group object in the categoryMathworldPlanetmath of schemes. Similarly, if S is a scheme, a 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 spaceMathworldPlanetmath do not form a group under the obvious choice for a group law.

We can view a group scheme G as a “group machineMathworldPlanetmath”: 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.

Title group scheme
Canonical name GroupScheme
Date of creation 2013-03-22 14:11:13
Last modified on 2013-03-22 14:11:13
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 4
Author archibal (4430)
Entry type Definition
Classification msc 14K99
Classification msc 14A15
Classification msc 14L10
Classification msc 20G15
Related topic Group
Related topic GroupVariety
Related topic Category
Related topic GroupObject
Related topic GroupSchemeOfMultiplicativeUnits
Related topic VarietyOfGroups