DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-12

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Morphisms of classical affine varieties

Definition

Let $X$ and $Y$ be classical affine varieties over an algebraically closed field $k$ . A map $φ : X \to Y$ is a morphism when for every open subset $V \subseteq Y$ and every regular function $f \in O_{Y} (V)$ , the pullback $f \circ φ : φ^{- 1} (V) \to k$ is regular on $φ^{- 1} (V)$ .

Because $Y$ is affine, it is enough to test pullback on global regular functions. By Global regular functions on a classical affine variety are its coordinate ring, those are exactly the elements of $k [Y]$ .

Depends on

Used by

The cusp parametrization t mapsto (t²,t³) is bijective but not an isomorphism Counterexample
Dominant morphisms and dominant rational maps Definition
Images and fibres of a regular map Definition
morphism to projective space homogeneous coordinates Definition
Projective classical morphisms Definition
Rational maps between irreducible classical affine varieties Definition
Reduced closed-point fibres and their dimension Definition
A polynomial map and its pullback on coordinate rings Example
Morphisms from an irreducible affine variety to an affine variety are determined by a dense open subset Lemma
Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms Theorem

Dependency tree · two levels

7 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, Proposition 3.26 (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, Exercise 3.2.5 (standard reference, not scraped)