Group scheme

Summary

A group scheme over a base scheme $S$ is a scheme over $S$ with multiplication, unit and inverse morphisms that satisfy the group axioms. Equivalently, its points with values in every $S$-scheme form a group, naturally in the $S$-scheme. Group schemes include algebraic groups over fields, such as $\mathrm{GL}_n$ and elliptic curves. They also include infinitesimal objects such as $\mu_p$ in characteristic $p$, which have no analogue among classical groups.