LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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 for the Zariski topology on an affine variety

Statement

Let $X$ be a classical affine variety over an algebraically closed field $k$ . Then the principal opens $D_{X} (\overline{f})$ form a basis for the Zariski topology on $X$ . In particular, $D_{X} (\overline{f}) \cap D_{X} (\overline{g}) = D_{X} (\overline{f} \overline{g})$ for all $\overline{f}, \overline{g} \in k [X]$ .

Facts & Assumptions

Given: A classical affine variety $X$ over an algebraically closed field $k$ .

[L1]

For $\overline{f} \in k [X]$ , the principal open is $D_{X} (\overline{f}) = {x \in X : \overline{f} (x) \neq 0}$ (A principal open subset of a classical affine variety).

[L2]

Closed subsets of affine space are zero loci of sets of polynomials (Zero loci in affine space are the closed sets of the classical Zariski topology).

Proof

technique · direct

1.1L1givenalgebra

For any $\overline{f} \in k [X]$ , the set $D_{X} (\overline{f})$ is open because it is the complement in $X$ of the closed subset where $\overline{f}$ vanishes. Also a point lies in $D_{X} (\overline{f}) \cap D_{X} (\overline{g})$ exactly when both $\overline{f}$ and $\overline{g}$ are nonzero there, equivalently when $\overline{f} \overline{g}$ is nonzero there. Thus $D_{X} (\overline{f}) \cap D_{X} (\overline{g}) = D_{X} (\overline{f} \overline{g})$ .

1.2L1L2choose

Let $U \subseteq X$ be Zariski-open and $x \in U$ . Then $X ∖ U$ is closed in $X$ , so there is a set $S$ of polynomials on the ambient affine space with $X ∖ U = X \cap V (S)$ by [L2]. Since $x \notin V (S)$ , choose $f \in S$ with $f (x) \neq 0$ . Then $x \in D_{X} (\overline{f}) \subseteq U$ . Hence every open set is a union of principal opens.

Principal opens form a basis for the Zariski topology on an affine variety

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources