---
tag: TJ04VQP4NB
kind: concept
name: Group scheme
msc: [14L15]
sources:
  - url: https://en.wikipedia.org/wiki/Group_scheme
same_as:
  wikidata: Q5611270
  wikipedia: 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.

## Explanations

### L1 human

A Lie group is a manifold that is also a group, with smooth operations.
A group scheme is the same idea in algebraic geometry: a scheme whose points can be multiplied by polynomial formulas.
The simplest example is the multiplicative group $\mathbb{G}_m$, whose points are the invertible numbers, multiplied as usual.
Over a general base, a group scheme is a family of such groups that varies with a parameter.