Algebraic group - Knowlpedia

Definition

An algebraic group over a field $k$ is a group scheme $G$ of finite type over $k$ . Thus $G$ has multiplication, identity, and inversion morphisms

m:G\times_kG\to G,\qquad e:\operatorname{Spec}k\to G,\qquad i:G\to G

satisfying the group axioms as identities of morphisms.

Linear and projective cases

An algebraic group is linear if its underlying scheme is affine. Equivalently, it admits a closed immersion into $GL_n$ for some $n$ . Projective algebraic groups, such as abelian varieties, need not be linear.

Points

For every $k$ -algebra $R$ , the set $G(R)$ is a group, functorially in $R$ . The abstract group $G(k)$ does not by itself determine the scheme-theoretic structure of $G$ .

References

T. A. Springer, Linear Algebraic Groups, 2nd ed., Birkhäuser, 1998. DOI.