CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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The affine algebraic-set dictionary is contravariantly full and faithful on classical affine varieties

Statement

Over an algebraically closed field $k$ , the object assignment $X ⟼ k [X]$ from affine algebraic sets to reduced affine $k$ -algebras restricts on classical affine varieties to a contravariant full and faithful functor.

Facts & Assumptions

Given: An algebraically closed field $k$ .

[L1]

Affine algebraic sets correspond to reduced affine $k$ -algebras at the object level (Affine algebraic sets and reduced affine k-algebras at the object level).

[L2]

Morphisms of classical affine varieties correspond contravariantly to $k$ -algebra homomorphisms of coordinate rings (Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms).

Proof

technique · direct

1.1

Fact [L1] supplies the object assignment $X \mapsto k [X]$ for affine algebraic sets.

L1given

2.1

Let $X$ and $Y$ be classical affine varieties. Fact [L2] gives a bijection $Mor (X, Y) ≅ {Hom}_{k -alg} (k [Y], k [X])$ that respects identities and composition in the contravariant order. Therefore, on the full subcategory of classical affine varieties, the assignment of step 1.1 is contravariantly full and faithful.

L2step 1.1

3.1

This is exactly the claimed morphism-level completion of the affine algebraic-set dictionary on classical affine varieties.

step 2.1∎

Depends on

Used by

The product of affine varieties has coordinate ring k[X] tensorₖ k[Y] Theorem

Dependency tree · two levels

9 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 3.25 (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, Theorem 3.2.1 together with the affine dictionary of Chapter 1 (standard reference, not scraped)