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A nontrivial smooth connected unipotent group with split torus action over a perfect field has a stable central G_a

Statement

Assume the Axiom of Choice inherited from the cited smoothness, quotient and reduction suppliers (The Axiom of Choice).

Let $k$ be a perfect field, let $U$ be a smooth connected unipotent algebraic group over $k$ (Trigonalizable algebraic groups), and let $T$ be a split torus acting on $U$ by group automorphisms (Groups of multiplicative type and tori). If $U \neq 1$ , there is a closed subgroup $N \subseteq U$ that is central in $U$ , stable under $T$ , and isomorphic to $G_{a}$ .

Facts & Assumptions

Given: AC, a perfect field $k$ , a smooth connected unipotent $k$ -group $U \neq 1$ , and a split torus $T$ acting on $U$ by group automorphisms.

[F1]

The semidirect product $H = U ⋊ T$ is a smooth connected trigonalizable affine group with largest normal unipotent subgroup $H_{u} = U$ : the quotient $H / U ≅ T$ is a torus, and a normal unipotent closed subgroup of $H$ maps into this torus, where it is trivial because a closed subgroup that is both unipotent and diagonalizable is trivial. (Trigonalizable algebraic groups, Groups of multiplicative type and tori, A subgroup that is both unipotent and diagonalizable is trivial)

[F2]

Assume AC. The smooth connected trigonalizable group $H$ has a normal series $H \supseteq H_{0} = U \supseteq H_{1} \supseteq \dots \supseteq H_{r} = 1$ in which every term $H_{i}$ with $i \geq 0$ is smooth, connected and normal in $H$ , and every successive quotient $H_{i} / H_{i + 1}$ is isomorphic to $G_{a}$ ; the series refines the $H / H_{u}$ -equivariant series with quotients embedded in $G_{a}$ . (Unipotent radicals of smooth connected trigonalizable groups over perfect fields have normal G_a series, Trigonalizable groups have a normal series with a multiplicative quotient and additive subgroup quotients)

[F3]

An automorphism of $G_{a}$ over a field is linear: an automorphism of the polynomial algebra has degree one, and preserving zero removes its constant term. For a smooth affine acting group $H$ , apply this fact only to its points over an algebraic closure. Smooth schemes have schematically dense rational points there, so coefficients vanishing at those points vanish in $O (H_{\overset{ˉ}{k}})$ and hence in $O (H)$ . This proves linearity of an $H$ -action on $G_{a}$ below; it does not identify the full automorphism functor with $G_{m}$ . (Rational points of smooth finite-type schemes over a separably closed field are schematically dense)

[F4]

Every nonzero rational representation of the unipotent group $U$ has a nonzero fixed vector; a vector fixed by $U$ in a representation that factors through a quotient of $U$ is fixed by that quotient; and the kernel of the standard action of $G_{m}$ on $A^{1}$ is trivial. (Unipotent algebraic groups and unipotent representations)

[A1]

The Axiom of Choice is inherited through the cited suppliers and is the axiom of The Axiom of Choice.

Proof

Given: AC, a perfect field $k$ , a smooth connected unipotent $k$ -group $U \neq 1$ , and a split torus $T$ acting on $U$ by group automorphisms.

1.1F1F2

Form $H = U ⋊ T$ , which is smooth connected trigonalizable with $H_{u} = U$ by [F1]; by [F2] fix a normal series $H \supseteq H_{0} = U \supseteq H_{1} \supseteq \dots \supseteq H_{r} = 1$ with each $H_{i}$ ( $i \geq 0$ ) smooth, connected and normal in $H$ and each quotient $H_{i} / H_{i + 1} ≅ G_{a}$ . Since $U \neq 1$ the series is nontrivial; let $N = H_{r - 1}$ be its last nontrivial term. Then $N \subseteq U$ , and $N$ is smooth, connected and normal in $H$ , hence stable under the conjugation action of $T$ ; since $H_{r} = 1$ , the last quotient is $N = N / H_{r} ≅ G_{a}$ .

2.1F3F4step 1.1algebra

Identify $N$ with $G_{a}$ and write the conjugation coaction as $x \mapsto \sum_{j \geq 0} a_{j} \otimes x^{j}$ , with $a_{j} \in O (H)$ . The constant coefficient is zero because the action fixes the identity. Over an algebraic closure, evaluation at every $h \in H (\overset{ˉ}{k})$ is a field-valued automorphism of $G_{a}$ , so $a_{j} (h) = 0$ for $j > 1$ . The smooth reduced group $H_{\overset{ˉ}{k}}$ has schematically dense rational points by [F3], hence every $a_{j}$ for $j > 1$ is zero. Inversion in $H$ supplies an inverse for $a_{1}$ , and the action law gives $Δ (a_{1}) = a_{1} \otimes a_{1}$ , so this coaction is scalar multiplication through a character $H \to G_{m}$ . Restricting it to $U$ gives a one-dimensional rational representation. By [F4] it has a nonzero invariant vector, so the entire line is invariant and the character of $U$ is trivial as a group-scheme morphism. Thus conjugation $U \times N \to N$ is trivial and $N$ is central in $U$ .

3.1A1step 1.1step 2.1∎

Collecting: $N \subseteq U$ is a closed subgroup isomorphic to $G_{a}$ , central in $U$ by [step 2.1] and stable under $T$ by [step 1.1], which is the required subgroup.

Depends on

Used by

Fixed loci and centralizers of torus actions are connected 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)