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.

Germs of regular functions and the local ring at a point of a classical affine variety

Definition

Let $X$ be a classical affine variety, let $x \in X$ , and let $f \in O_{X} (U)$ and $g \in O_{X} (V)$ be regular functions on open neighbourhoods $U, V$ of $x$ . The two pairs $(U, f)$ and $(V, g)$ are equivalent at $x$ when there exists an open neighbourhood $W \subseteq U \cap V$ of $x$ such that $f ∣_{W} = g ∣_{W}$ .

An equivalence class is called the germ of a regular function at $x$ and is written $f_{x}$ when $f$ represents it.

The set of all such germs is a ring under representativewise addition and multiplication. It is the local ring of $X$ at $x$ and is denoted $O_{X, x} .$

Depends on

Regular functions on open subsets of a classical affine variety

Used by

Local intersection multiplicity of two plane curves Definition
Multiplicity of a plane curve at a point Definition
Normal points and normal varieties Definition
Uniformising parameters at smooth points of a plane curve Definition
Multiplicity one characterises smooth points with a unique tangent Lemma
The local ring at a point of an affine variety is the localization at its maximal ideal Theorem

Dependency tree · two levels

3 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, Chapter 3c (standard reference, not scraped)
Michael Artin, Notes for a Course in Algebraic Geometry, 5.1.10 (standard reference, not scraped)