LemmaStatement: 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.

Properness over a field can be checked after field extension

Statement

Assume AC. Let $X$ be a separated finite-type scheme over a field $k$ , and let $K / k$ be any field extension. Then $X$ is proper over $k$ if and only if $X_{K}$ is proper over $K$ . In particular this can be checked over an algebraic closure, and applies to completeness of group varieties, where complete means proper. No algebraicity or separability of $K / k$ is required.

Facts & Assumptions

[F1]

Properness is separatedness, finite type and universal closedness, and is preserved and reflected by fpqc base change under AC. (Proper morphisms, Properness descends through fpqc base change)

Proof

Given: AC, $X / k$ , and a field extension $K / k$ as stated.

1.1givenconstructalgebra

The map $Spec K \to Spec k$ is flat, because every vector space over a field is flat; it is surjective because both spectra have one point; and it is quasi-compact because it is affine. Thus it is an fpqc covering morphism. Its pullback of $X \to Spec k$ is precisely $X_{K} \to Spec K$ . These facts hold for infinite and inseparable extensions too.

2.1F1step 1.1∎

Apply the fpqc properness equivalence [F1] to this covering. It gives both implications in the statement. Taking $K$ to be an algebraic closure gives the geometric test, and the same equivalence says complete group varieties descend and ascend, with complete interpreted as proper. AC is used only through [F1].

Depends on

Used by

Dependency tree · two levels

11 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), proof of Theorem 8.26 and Appendix A.75, p.154 (standard reference, not scraped)
Brion, Some structure theorems for algebraic groups, Sections 4.2-4.3 (standard reference, not scraped)