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 regular function on an open subset of a classical affine variety

Definition

Let $U$ be open in an affine algebraic set $X$ and $A = k [X]$ . A function $s : U \to k$ is regular if for each $x \in U$ there are an open neighbourhood $W \subseteq U$ of $x$ and $g, h \in A$ with $h (y) \neq 0$ for every $y \in W$ such that $s (y) = g (y) / h (y)$ there. Write $O_{X} (U)$ for these functions. The quotient need only hold locally; no single fraction on all of $U$ is required. On the empty open the unique empty function is regular. Constants are regular by $c = c / 1$ .

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, Definition 3.8, p. 61. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

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

J. S. Milne, Algebraic Geometry v6.10, Definition 3.8, p. 61 (standard reference, not scraped)