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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A proper geometrically connected group variety is commutative

Statement

Assume the Axiom of Choice. Every abelian variety over a field is commutative. More generally, if $A$ is an abelian variety and $H$ a connected separated finite-type $k$ -group scheme, every group homomorphism $u : A \to H$ has central image, scheme theoretically.

Facts & Assumptions

[F1]

Abelian varieties are proper geometrically integral group varieties, with a rational identity. (Abelian varieties over a field)

[F2]

A morphism from $X \times Y$ with $X$ proper geometrically integral and rationally pointed, $Y$ connected, and separated target, constant on a rational fibre, factors through $Y$ , under AC. (Rigidity for a proper geometrically integral factor)

Proof

Given: AC, an abelian variety $A$ , a connected finite-type group scheme $H$ , and a homomorphism $u : A \to H$ .

1.1F1F2givenalgebra

Form the morphism $c : A \times_{k} H \to H$ given on scheme-valued points by $c (a, h) = u (a) h u (a)^{- 1} h^{- 1}$ . Group multiplication and inverse show that it is a scheme morphism. On $A \times {e_{H}}$ it is identically $e_{H}$ . Apply [F2] with $X = A$ , $Y = H$ , and $Z = H$ ; algebraic group schemes here are separated. Therefore $c (a, h) = c (e_{A}, h) = e_{H}$ as an identity of scheme morphisms. Equivalently, conjugation of $u (a)$ by every scheme-valued point of $H$ fixes it. This is precisely the statement that $u$ factors through the scheme-theoretic centre.

2.1step 1.1given∎

Take $H = A$ and $u = {id}_{A}$ in step 1.1. The equality $a h a^{- 1} h^{- 1} = e$ then gives $a h = h a$ on every $k$ -scheme of points and hence as a morphism identity. Thus the group law of $A$ is commutative. AC is used through [F2].

Depends on

Used by

Cited to discharge well-definedness by Abelian varieties over a field.

Dependency tree · two levels

16 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

Milne, Algebraic Groups (2022), 8.13 and 8.20 (standard reference, not scraped)
Brion, Some structure theorems for algebraic groups, Corollary 3.1.7 (standard reference, not scraped)