Definition Definition of Linear Algebraic Groups
definition needs_definition_review

Definition of Linear Algebraic Groups

Setup

Let $k$ be a field. Write $\operatorname{Sch}_{/k}$ for schemes over $\operatorname{Spec} k$, and write $\operatorname{AffSch}_{/k}$ for affine schemes over $\operatorname{Spec} k$.

A group object in $\operatorname{Sch}_{/k}$ is a $k$-scheme $G$ equipped with morphisms

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

satisfying the associativity, identity, and inverse diagrams.

Definition

A linear algebraic group over $k$ is an affine group scheme of finite type over $k$. Equivalently, it is a group object $G$ in $\operatorname{Sch}_{/k}$ such that the structure morphism

$$ G\to \operatorname{Spec} k $$

is affine and of finite type.

Equivalently, it is a group object in $\operatorname{AffSch}_{/k}$ whose coordinate ring is a finitely generated $k$-algebra.

Coordinate Ring Form

If $G=\operatorname{Spec} A$ is affine, then the group structure on $G$ is the same as a commutative Hopf $k$-algebra structure on $A=k[G]$:

$$ \Delta:A\to A\otimes_k A,\qquad \epsilon:A\to k,\qquad S:A\to A. $$

Here:

  • $\Delta$ is dual to multiplication $m:G\times_k G\to G$;
  • $\epsilon$ is dual to the identity section $e:\operatorname{Spec}k\to G$;
  • $S$ is dual to inversion $i:G\to G$.

Thus a linear algebraic group over $k$ is equivalently the spectrum of a finitely generated commutative Hopf $k$-algebra.

Relation With Classical Language

When $k$ is algebraically closed and one works only with reduced objects, this agrees with the classical language of affine algebraic groups: affine algebraic varieties with regular multiplication and inverse maps.

The scheme-theoretic definition is slightly broader. In positive characteristic it allows nonreduced examples, such as infinitesimal group schemes. If a text intends only smooth groups, smoothness should be added as a separate hypothesis:

$$ G\to \operatorname{Spec} k \text{ is smooth.} $$

Nonexamples

Not every group object over $\operatorname{Spec} k$ is linear algebraic. For example, an elliptic curve over $k$ is a group scheme of finite type over $k$, but it is proper rather than affine, so it is not a linear algebraic group.

This is why the word affine is essential in the definition.

Examples

  • The additive group $\mathbb{G}_a=\operatorname{Spec} k[x]$.
  • The multiplicative group $\mathbb{G}_m=\operatorname{Spec} k[x,x^{-1}]$.
  • The general linear group $\operatorname{GL}_n$.
  • The special linear group $\operatorname{SL}_n$.
  • Any closed subgroup scheme of $\operatorname{GL}_n$.

Lean Formalization Comment

In Lean/mathlib terms, the natural package for this definition is:

G : Over (Spec (.of k))
[CategoryTheory.GrpObj G]
[AlgebraicGeometry.IsAffineHom G.hom]
[AlgebraicGeometry.LocallyOfFiniteType G.hom]

Smoothness should be an additional typeclass assumption when the intended convention requires smooth algebraic groups.

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