Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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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.

Affine schemes are contravariantly equivalent to commutative rings

Statement

For commutative unital rings A,B, the assignment φ↦Spec⁡(φ) gives a natural bijection Hom⁡CRing(A,B)≅Hom⁡LRS(Spec⁡B,Spec⁡A). Consequently A↦Spec⁡A is a contravariant equivalence from commutative rings to affine schemes, with quasi-inverse global sections.

Facts & Assumptions

Given: Commutative unital rings A,B and a morphism u:Spec⁡B→Spec⁡A.

[F1]

Global sections of an affine spectrum recover its ring (Global functions on Spec A recover A).

[F2]

A ring map gives a morphism of affine spectra (The map of affine spectra induced by a ring homomorphism), and its stalk maps are local (The stalk maps induced by a ring map are local).

Proof

technique · direct
1.1F2

By [F2], every ring map A→B induces a locally ringed-space morphism Spec⁡B→Spec⁡A.

1.2F1

A morphism u gives a global-sections map u♯:A→B using [F1].

2.1step 1.2algebra

Locality determines its point map by contraction and localization determines each basic-open section map, so u=Spec⁡(u♯).

3.1step 1.1step 2.1∎

The constructions of steps 1.1--2.1 are inverse and natural; global sections is the quasi-inverse.

Depends on

Dependency tree · 2 levels

Sources