LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Simple rational representations are finite-dimensional

Statement

Let $G$ be an affine group scheme of finite type over a field $k$ . Then every simple rational representation $(V, r)$ of $G$ is finite-dimensional (Rational representations and comodules of an affine group scheme, Simple and semisimple rational representations).

Facts & Assumptions

Given: An affine group scheme $G$ of finite type over $k$ with coordinate Hopf algebra $A = O (G)$ , and a simple rational representation $(V, r)$ , so $V \neq 0$ and the only subrepresentations of $V$ are $0$ and $V$ (Simple and semisimple rational representations).

[F1]

Finite-dimensional subcomodules. For every finite subset $S$ of an $A$ -comodule $M$ there is a finite-dimensional subcomodule $N \subseteq M$ containing $S$ ; consequently $M$ is the directed union of its finite-dimensional subcomodules (Every element of a comodule lies in a finite-dimensional subcomodule).

[F2]

Subrepresentations are subcomodules. Under the comodule dictionary, subrepresentations of $V$ correspond to subcomodules, and a nonzero subcomodule is a nonzero subrepresentation (Rational representations and comodules of an affine group scheme).

Proof

technique · direct

1.1given

Since $V \neq 0$ , choose a nonzero vector $v \in V$ .

2.1F1step 1.1

By [F1] there is a finite-dimensional subcomodule $W \subseteq V$ containing $v$ .

3.1F2givenstep 2.1∎

By [F2] the subspace $W$ is a subrepresentation of $V$ ; it is nonzero because $v \in W$ . Since $V$ is simple, its only subrepresentations are $0$ and $V$ , so $W = V$ . Hence $V$ is finite-dimensional, as claimed.

Remarks

The only input is Milne 4.8: the comodule structure makes every element lie in a finite-dimensional subcomodule, and a simple module cannot have a nonzero proper submodule.
No hypothesis on $k$ beyond being a field is used, and no choice principle: the finite-dimensional subcomodule is produced from finitely many coefficients of $ρ (v)$ .

Depends on

Used by

Dependency tree · two levels

17 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)
Robert Steinberg, Lectures on Chevalley Groups (Yale University, 1967; notes prepared by J. Faulkner and R. Wilson) (standard reference, not scraped)