# group scheme

A group scheme is a group object in the category 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 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 $S$.

 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