DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

A principal open subset of a classical affine variety

Definition

Let $X$ be a classical affine variety over an algebraically closed field $k$ , and let $\overline{f} \in k [X]$ . The principal open subset determined by $\overline{f}$ is $D_{X} (\overline{f}) : = {x \in X : \overline{f} (x) \neq 0} .$

If $f \in k [x_{1}, \dots, x_{n}]$ is any representative of $\overline{f}$ , then $D_{X} (\overline{f})$ depends only on the class $\overline{f}$ , not on the chosen representative. Its boundary cases are $D_{X} (0) = \emptyset and D_{X} (1) = X .$

Depends on

Used by

Dependency tree · two levels

5 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, Definition 3.8 (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, localization discussion after Lemma 1.6.3 (standard reference, not scraped)