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.

Principal opens form a basis and multiply under intersection

Statement

The principal opens $D_{X} (f)$ form an open basis on every affine algebraic set $X$ , and $D_{X} (f) \cap D_{X} (g) = D_{X} (f g)$ .

Facts & Assumptions

Given: An affine algebraic set $X \subseteq k^{n}$ over algebraically closed $k$ , elements $f, g \in k [X]$ , and an open $U \subseteq X$ .

[F1]

A closed subset of X is given by simultaneous polynomial equations (Classical affine zero loci form the Zariski closed sets).

[F2]

D(f) is the set where f is nonzero (A principal open subset of a classical affine variety).

Proof

technique · direct

1.1F2givenalgebra

For $f, g \in k [X]$ , $f (x) g (x) \neq 0$ in the field $k$ exactly when both factors are nonzero. This proves the intersection identity, including $f = 0$ or $g = 0$ and $f = 1$ .

2.1F1F2given∎

If $U = X ∖ (X \cap V (S))$ is open, then $x \in U$ exactly when some $s \in S$ does not vanish at $x$ . Hence $U = ⋃_{s \in S} D_{X} (\overset{ˉ}{s})$ . Each member is open and contained in $U$ . This gives a principal neighbourhood of each point of $U$ , and the empty $S$ gives the empty union.

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, Proposition 2.37, p. 49. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

Dependency tree · 2 levels

Sources

J. S. Milne, Algebraic Geometry v6.10, Proposition 2.37, p. 49