DefinitionDefinition: AI-adaptedProof: 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.

Affine open subsets of a classical affine variety

Definition

Let $X$ be a classical affine variety. On this page, an affine open subset of $X$ means a principal open subset $U = D_{X} (\overline{f})$ for some $\overline{f} \in k [X]$ .

Its coordinate ring is $k [U] : = O_{X} (U),$ and Regular functions on a principal open are the principal localization of the coordinate ring identifies it with the principal localisation $k [X]_{\overline{f}}$ .

This page uses the classical basic-affine opens only. Later pages may enlarge the phrase "affine open" to the intrinsic scheme-theoretic notion.

Depends on

Used by

Dependency tree · two levels

10 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

Michael Artin, Notes for a Course in Algebraic Geometry, Proposition 2.6.1 (standard reference, not scraped)
J. S. Milne, Algebraic Geometry, Chapter 3c (standard reference, not scraped)