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.

Regular functions on open subsets of a classical affine variety

Definition

Let $X$ be a classical affine variety over an algebraically closed field $k$ , and let $U \subseteq X$ be open. A function $f : U \to k$ is regular on $U$ when for every point $x \in U$ there exist an open neighbourhood $W \subseteq U$ of $x$ and elements $a, b \in k [X]$ such that $b (y) \neq 0$ for every $y \in W$ and $f (y) = \frac{a (y)}{b (y)} for every y \in W .$

The set of regular functions on $U$ is denoted $O_{X} (U)$ .

Thus a regular function is locally a quotient of global regular functions from the affine variety itself, with a denominator that stays nonzero on the neighbourhood where that formula is used.

Regular functions on open subsets of a classical affine variety

Definition

Depends on

Dependency tree · 2 levels

Sources