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.
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A proper geometrically integral affine scheme is a point

Statement

Assume the Axiom of Choice. A proper geometrically integral affine finite-type $k$ -scheme is $Spec k$ . Every morphism from a proper geometrically integral finite-type $k$ -scheme to an affine $k$ -scheme factors through a $k$ -rational point. In particular a positive-dimensional abelian variety is not affine.

Facts & Assumptions

[F1]

Under AC, $Γ (X, O_{X}) = k$ for proper geometrically integral $X$ . (Global functions on proper integral schemes form a finite extension of the base field)

[F2]

Global sections recover the ring of an affine scheme, and morphisms into an affine scheme correspond to ring maps on global sections. (Global functions on Spec A recover A, Morphisms to an affine scheme and global sections)

[F3]

An abelian variety is proper and geometrically integral. (Abelian varieties over a field)

Proof

Given: AC and $X$ proper geometrically integral of finite type over $k$ .

1.1F1F2given

If $X = Spec B$ is affine, [F1] and [F2] identify $B$ with $k$ as a $k$ -algebra. Taking spectra gives $X ≅ Spec k$ .

2.1F1F2F3step 1.1∎

For an arbitrary affine target $T = Spec B$ , a $k$ -morphism $X \to T$ corresponds by [F2] to a $k$ -algebra map $B \to Γ (X, O_{X}) = k$ . That map defines a $k$ -rational point of $T$ , and naturality in [F2] gives the desired factorization through $X \to Spec k$ . By [F3], an affine abelian variety would be a point by step 1.1, so a positive-dimensional abelian variety cannot be affine. AC is used precisely through [F1].

Depends on

Used by

Dependency tree · two levels

37 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), Example 8.4 and Appendix A.75 (standard reference, not scraped)
Stacks Project, Varieties, Lemma 33.9.3 (standard reference, not scraped)