LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

Every nonempty open of a classical affine variety is dense

Statement

Every nonempty open subset of an irreducible classical affine variety $X$ is dense. Every finite intersection of nonempty open subsets of $X$ is nonempty and dense; the intersection of the empty family is $X$ .

Facts & Assumptions

Given: An affine variety $X$ over algebraically closed $k$ , a nonempty open of $X$ , and a finite family of nonempty opens of $X$ .

[F1]

Nonempty opens of an irreducible space are dense and irreducible, and two such opens meet (Irreducibility is equivalent to the nonempty-open intersection criterion).

Proof

technique · direct

1.1

The density assertion is F1 with the irreducible nonempty space $X$ . For a finite list $U_{1}, \dots, U_{r}$ , start with $W_{0} = X$ and set $W_{j} = W_{j - 1} \cap U_{j}$ . If $W_{j - 1}$ is nonempty open, F1 gives $W_{j} \neq \emptyset$ ; intersection preserves openness. Finite induction gives $W_{r}$ nonempty open.

F1given

2.1

F1 now makes $W_{r}$ dense. This includes $r = 0$ , since $W_{0} = X$ is nonempty and its closure is itself.

F1step 1.1∎

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, §2h p. 45. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

Irreducibility is equivalent to the nonempty-open intersection criterion

Used by

Dependency tree · two levels

2 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 Geometry v6.10, §2h p. 45 (standard reference, not scraped)