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Extensions of multiplicative-type groups by a one-dimensional vector group with a linear action split

Statement

Assume the Axiom of Choice. Let $k$ be a field, let $G$ be an affine algebraic group of multiplicative type over $k$ (Groups of multiplicative type and tori, Affine schemes and their coordinate rings), and let $0 \to U \to E \to G \to 1$ be an extension of affine algebraic groups in which $U ≅ G_{a}$ and the induced action of $G$ on $U$ is linear (equivalently, $U$ is identified with $G_{a}$ on which $G$ acts through a character). Then the extension splits: $E ≅ U ⋊ G$ , and the projection admits a homomorphism of algebraic groups as a section.

Facts & Assumptions

Given: The Axiom of Choice, a field $k$ , an affine group $G$ of multiplicative type, and an extension $0 \to U \to E \to G \to 1$ with $U ≅ G_{a}$ and linear $G$ -action.

[F1]

For an extension of group functors $0 \to M \to E \to G \to 1$ that admits a section as a map of set-valued functors (a Hochschild extension), the induced conjugation action of $G$ on $M$ is defined and the equivalence classes of such extensions inducing that action are classified by $H^{2} (G, M)$ ; the extension is trivial, i.e. $E ≅ M ⋊ G$ , exactly when the class vanishes. (Crossed homomorphisms, principal crossed homomorphisms and Hochschild extensions, Hochschild cohomology of algebraic groups and the classification of Hochschild extensions)

[F2]

If $U ≅ G_{a}$ is a commutative group scheme with a $G$ -action, the projection $E \to G$ makes $E$ a $U$ -torsor over the affine base $G$ ; every $G_{a}$ -torsor over an affine scheme is trivial, so the projection admits a scheme section and $E$ is a Hochschild extension in the sense of [F1]. (Normal subgroup quotients of finite-type group schemes exist as fppf scheme quotients, Torsors under the additive group over an affine scheme are trivial, Crossed homomorphisms, principal crossed homomorphisms and Hochschild extensions)

[F3]

A group of multiplicative type over a field is linearly reductive, and a linearly reductive affine algebraic group has $H^{n} (G, V) = 0$ for all $n \geq 1$ and every rational representation $V$ . (Groups of multiplicative type are linearly reductive, Higher Hochschild cohomology vanishes for linearly reductive groups)

Proof

Given: The Axiom of Choice, a field $k$ , an affine group $G$ of multiplicative type, and an extension $0 \to U \to E \to G \to 1$ with $U ≅ G_{a}$ and linear $G$ -action.

1.1F1F2

The action of $G$ on $U$ is linear, so it endows $U ≅ G_{a}$ with the structure of a rational representation of $G$ ; the projection $E \to G$ is a torsor under this group scheme $U$ over the affine base $G$ , and by [F2] it is trivial, so there is a morphism of schemes $s : G \to E$ with $π \circ s = {id}_{G}$ . Thus the extension is a Hochschild extension, and by [F1] its isomorphism class corresponds to an element of $H^{2} (G, U)$ with coefficients in the representation $U$ of $G$ .

2.1F1F3step 1.1∎

Since $G$ is of multiplicative type, [F3] makes it linearly reductive, and therefore $H^{2} (G, U) = 0$ for the rational representation $U$ ; by [F1] the class of the extension vanishes, so the extension is equivalent to the split extension $U ⋊ G$ and is therefore split by a homomorphism of algebraic groups.

Depends on

Used by

Splitting trigonalizable extensions: algebraically closed fields and two perfect-field cases Theorem

Dependency tree · two levels

54 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)
J. S. Milne, Algebraic Groups (v2.00, 20 December 2015 author-hosted preliminary edition) (standard reference, not scraped)