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One-dimensional smooth connected affine groups over perfect fields are additive groups or tori

Statement

Assume the Axiom of Choice inherited from the torus-splitting and geometric suppliers (The Axiom of Choice).

Let $k$ be a perfect field and let $G$ be a smooth connected affine algebraic group of dimension one over $k$ (Affine schemes and their coordinate rings, Smooth morphism of schemes, Affine schemes and their coordinate rings). Then either $G$ becomes isomorphic to $G_{m}$ over a finite separable extension of $k$ , or it becomes isomorphic to $G_{a}$ over a finite purely inseparable extension of $k$ . Over an algebraically closed field, $G_{a}$ and $G_{m}$ are the only connected affine group varieties of dimension one; an elliptic curve shows that affineness cannot be dropped.

Facts & Assumptions

Given: AC, a perfect field $k$ and a smooth connected affine algebraic group $G$ of dimension one over $k$ .

[F1]

A smooth connected algebraic group of dimension $1$ is commutative; more generally the commutative structure theory provides, for a smooth connected commutative affine group $H$ , a largest subgroup $H_{s}$ of multiplicative type and a largest unipotent subgroup $H_{u}$ , with $H / H_{s}$ unipotent, $H / H_{u}$ of multiplicative type, and $\dim H = \dim H_{s} + \dim H_{u}$ . Over a perfect field the commutative structure theorem gives $H ≅ H_{s} \times H_{u}$ , and when $H$ is smooth connected both factors are smooth connected (Milne Theorem 16.13(b) and Corollary 16.15). (Milne, Algebraic Groups, Proposition 14.25 and Sections 16.13-16.15; the local library develops the multiplicative-type dictionary in Groups of multiplicative type and tori but not this structure theorem.)

[F2]

A smooth connected unipotent group of dimension one over an algebraically closed field is isomorphic to $G_{a}$ ; over a perfect field such a group becomes isomorphic to $G_{a}$ over a finite purely inseparable extension. (Milne, Algebraic Groups, Corollary 14.53 and Corollary 16.16; with Unipotent groups are exactly the subgroups of some U_n, equivalently the groups with coconnected coordinate Hopf algebra for the identification of unipotence.)

[F3]

Assuming AC through the Galois character-module supplier, a one-dimensional torus over $k$ is split by a finite separable extension, and after splitting is isomorphic to $G_{m}$ ; the character module of a torus is a free abelian group of finite rank. (Tori correspond exactly to torsion-free character lattices, Groups of multiplicative type and tori)

[F4]

Assuming AC, a connected finite-type group scheme is geometrically connected. Thus a smooth connected zero-dimensional group has a single reduced geometric point and is identified with the trivial group by its identity section. (Connected finite-type groups are geometrically connected)

Proof

Given: AC, a perfect field $k$ and a smooth connected affine group $G$ of dimension one over $k$ .

1.1F1F4

By [F1] the group is commutative and, since $k$ is perfect, has the product decomposition $G ≅ G_{s} \times G_{u}$ into smooth connected multiplicative-type and unipotent factors. Their dimensions add to one, so one has dimension zero. A smooth connected zero-dimensional group over $k$ is trivial: it is geometrically connected by the group identity-component property, and finite étale, so its geometric fibre is a single reduced point and its identity section identifies it with $Spec k$ . Hence either $G = G_{s}$ or $G = G_{u}$ . This does not assert that arbitrary zero-dimensional unipotent group schemes are trivial.

2.1F3step 1.1

If $G = G_{s}$ , then $G$ is a smooth connected one-dimensional group of multiplicative type, hence a one-dimensional torus: its character module is free of rank one by [F3], and a one-dimensional torus is split by a finite separable extension, over which it becomes $G_{m}$ . This is the first alternative of the statement.

2.2F2step 1.1

If $G = G_{u}$ , then $G$ is smooth, connected, unipotent and one-dimensional; by [F2] it is isomorphic to $G_{a}$ over an algebraic closure, and over the perfect field $k$ it becomes isomorphic to $G_{a}$ over a finite purely inseparable extension. This is the second alternative.

3.1step 2.1step 2.2∎

Together, [step 2.1] and [step 2.2] prove that every smooth connected affine one-dimensional group is of one of the two described forms, and over an algebraically closed field the two alternatives read $G ≅ G_{m}$ or $G ≅ G_{a}$ . An elliptic curve $E$ over $k$ is a smooth connected group of dimension one that is proper and not affine, so it is not covered by the alternatives; this shows affineness is needed.

Depends on

Used by

Unipotent radicals of smooth connected trigonalizable groups over perfect fields have normal Gₐ series Lemma

Dependency tree · two levels

43 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Florian Herzig, Linear Algebraic Groups (University of Toronto lecture notes, 2013) (standard reference, not scraped)