group schemeGroupScheme2004-02-24 02:18:412004-02-24 02:18:41Definition% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newtheorem{theorem}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}A \emph{group scheme} is a group object in the category of schemes. Similarly, if $S$ is a scheme, a \emph{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 $S$.