DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-12

How statement and proof provenance work

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A rational function is regular at a point exactly when it lies in the local ring

Definition

Let $X$ be a classical affine variety, let $x \in X$ , and let $r \in k (X)$ . Under the identification $O_{X, x} ≅ k [X]_{m_{x}} \subseteq k (X)$ from The local ring at a point of an affine variety is the localization at its maximal ideal, the rational function $r$ is regular at $x$ when $r \in O_{X, x}$ .

If $r$ is regular at $x$ , its image in the residue field $κ (x) = O_{X, x} / m_{x}$ is called the value of $r$ at $x$ . In the classical setting The residue field at a classical point identifies this value with an element of the ground field $k$ .

Depends on

Used by

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Sources

Michael Artin, Notes for a Course in Algebraic Geometry, Lemma 3.4.4 (standard reference, not scraped)
J. S. Milne, Algebraic Geometry, Chapter 5k-l (standard reference, not scraped)