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Trigonalizable groups have a normal series with a multiplicative quotient and additive subgroup quotients

Statement

Assume the Axiom of Choice inherited from the faithful flag embedding and exact group-quotient suppliers (The Axiom of Choice).

Let $k$ be a field and let $G$ be a trigonalizable affine algebraic group over $k$ (Trigonalizable algebraic groups, Affine schemes and their coordinate rings, Group schemes of finite type over a field). Then there is a normal series $G \supseteq G_{0} \supseteq G_{1} \supseteq \dots \supseteq G_{r} = 1$ such that $G_{0} = G_{u}$ is the largest normal unipotent subgroup of $G$ , the quotient $G / G_{u}$ is of multiplicative type (Groups of multiplicative type and tori), and for each $i$ the quotient $G_{i} / G_{i + 1}$ embeds $G / G_{u}$ -equivariantly into $G_{a}$ with a linear action of $G / G_{u}$ (an action through the natural action of $G_{m}$ on $G_{a}$ ).

Facts & Assumptions

Given: AC, a field $k$ and a trigonalizable affine algebraic group $G$ over $k$ .

[F1]

By the flag criterion for trigonalizability, $G$ is isomorphic to a closed subgroup scheme of some upper triangular group $T_{n} = D_{n} ⋉ U_{n}$ ; in particular $G \subseteq T_{n}$ , $G / G_{u}$ embeds into $D_{n}$ , and $G_{u}$ will be defined as $G \cap U_{n}$ . (Trigonalizable groups, invariant flags and embeddings into T_n, The upper unitriangular group scheme U_n and its coordinate ring)

[F2]

The group $U_{n}$ has a central series $U_{n} = U_{n}^{(0)} \supseteq U_{n}^{(1)} \supseteq \dots \supseteq U_{n}^{(m)} = 1$ of closed subgroup schemes stable under conjugation by $T_{n}$ , with successive quotients canonically isomorphic to $G_{a}$ and with $D_{n}$ acting on each quotient through the character $d \mapsto d_{i} d_{j}^{- 1}$ . (The central series of U_n with additive quotients)

[F3]

A closed subgroup of the unipotent group $U_{n}$ is unipotent; the intersection of a unipotent closed subgroup with the diagonalizable group $D_{n}$ is trivial, and any normal unipotent closed subgroup $V \subseteq G$ maps into $D_{n} ≅ T_{n} / U_{n}$ , hence has trivial image and lies in $G \cap U_{n}$ . (Unipotent groups are exactly the subgroups of some U_n, equivalently the groups with coconnected coordinate Hopf algebra, A subgroup that is both unipotent and diagonalizable is trivial, Diagonalizable groups and their character modules)

[F4]

Assume AC. A homomorphism of finite-type group schemes has closed scheme-theoretic image isomorphic to its fppf quotient by the scheme kernel. Thus a trivial kernel makes it a closed immersion, and intersections compute kernels of restricted homomorphisms. (Group images are exact kernel quotients and preserve affine smooth connected properties)

[A1]

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

Proof

Given: AC, a field $k$ and a trigonalizable affine algebraic group $G$ over $k$ .

1.1F1F3F4

By [F1] fix a closed embedding $G \subseteq T_{n} = D_{n} ⋉ U_{n}$ and put $G_{u} = G \cap U_{n}$ . Then $G_{u}$ is a closed unipotent subgroup of $G$ by [F3], normal in $G$ because $U_{n}$ is normal in $T_{n}$ , and the map $G \to D_{n} ≅ T_{n} / U_{n}$ has scheme kernel $G_{u}$ , so [F4] identifies $G / G_{u}$ with its closed image there, hence is diagonalizable and therefore of multiplicative type. If $V \subseteq G$ is any normal unipotent closed subgroup, its image in $G / G_{u} \subseteq D_{n}$ is unipotent (as a quotient of a unipotent group) and diagonalizable, hence trivial by [F3], so $V \subseteq G_{u}$ : $G_{u}$ is the largest normal unipotent subgroup.

2.1F2F4step 1.1

Intersect the central series of $U_{n}$ from [F2] with $G$ : put $G_{i} = G \cap U_{n}^{(i)}$ for $0 \leq i \leq m$ . These are closed subgroup schemes of $G$ with $G_{0} = G \cap U_{n}$ , which is $G_{u}$ , and $G_{m} = 1$ ; each $G_{i}$ is normal in $G$ because $U_{n}^{(i)}$ is stable under $T_{n}$ . The restricted map $G_{i} \to U_{n}^{(i)} / U_{n}^{(i + 1)} ≅ G_{a}$ has scheme kernel $G_{i} \cap U_{n}^{(i + 1)} = G_{i + 1}$ . Thus [F4] identifies $G_{i} / G_{i + 1}$ with its closed scheme-theoretic image in $G_{a}$ , and this embedding is equivariant for the action of $G / G_{u} \subseteq D_{n}$ , which acts on the quotient through the linear character of [F2].

3.1A1step 1.1step 2.1∎

Dropping repeated terms from $G_{0} \supseteq G_{1} \supseteq \dots \supseteq G_{m} = 1$ yields the required normal series with $G_{0} = G_{u}$ , $G / G_{u}$ of multiplicative type, and each successive quotient $G_{i} / G_{i + 1}$ embedded $G / G_{u}$ -equivariantly into $G_{a}$ with a linear action; the series terminates at $1$ by [step 2.1]. This proves the theorem.

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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)