Algebraic group

Builds on Group scheme Morphism of finite type

Definition

Let $k$ be a field. An algebraic group over $k$ is a group scheme $G$ over $k$ whose underlying scheme is of finite type over $k$ . So $G$ comes with morphisms $m \colon G \times_k G \to G$ , $i \colon G \to G$ and $e \colon \operatorname{Spec} k \to G$ . They satisfy associativity, the unit law and the inverse law as identities of morphisms.

Remarks

Many authors also require $G$ to be smooth, or to be a variety; such groups are called group varieties. Over a field of characteristic $0$ , every algebraic group is smooth.

Example: the general linear group $\mathrm{GL}_n = \operatorname{Spec} k[x_{ij}, 1/\det]$ , with matrix multiplication and the inverse matrix. The entries of a product and of an inverse are polynomials in the $x_{ij}$ and $1/\det$ , so these are morphisms, and the group axioms hold.

Non-example: the constant group scheme $\Z_k = \coprod_{n \in \Z} \operatorname{Spec} k$ is a group scheme over $k$ . It has infinitely many connected components, so it is not of finite type over $k$ .

The group of points $G(k)$ is an abstract group, and $T \mapsto G(T)$ is a functor from $k$ -schemes to groups.

Reading path

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Tag	`HVMYKZ2PB5`
URI	`m2://HVMYKZ2PB5`
Kind	definition
Writing	W0 stub
MSC	14L10, 14L15, 20G15
Defines	Algebraic group
Uses	Group scheme, Morphism of finite type

Algebraic group

Definition

Remarks

Reading path

Further reading

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