DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

A principal open subset of a classical affine variety

Definition

For an affine algebraic set $X$ and $f \in k [X]$ , the principal open is $D_{X} (f) = {x \in X : f (x) \neq 0} .$ Evaluation is independent of a polynomial representative by Polynomial functions on an affine algebraic set are its coordinate ring. The complement is the relatively closed zero locus of that representative. In particular $D_{X} (0) = \emptyset$ and $D_{X} (1) = X$ . The notation applies to reducible and empty algebraic sets as well as varieties.

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, §2i p. 49. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

Used by

Principal opens form a basis and multiply under intersection Lemma

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

J. S. Milne, Algebraic Geometry v6.10, §2i p. 49 (standard reference, not scraped)