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

1.1L1given

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

2.1L2step 1.1

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.

3.1step 2.1∎

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

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources