Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
How statement and proof provenance work

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-scheme isomorphisms are exactly coordinate-ring isomorphisms in reverse direction

Statement

An affine-scheme morphism Spec⁡B→Spec⁡A is an isomorphism if and only if its associated homomorphism A→B is an isomorphism.

Facts & Assumptions

Given: A morphism u:Spec⁡B→Spec⁡A.

[F1]

Affine spectra and commutative rings are contravariantly equivalent (Affine schemes are contravariantly equivalent to commutative rings).

Proof

technique · direct
1.1F1

If u is an isomorphism, [F1] carries its inverse to an inverse of the associated ring map A→B.

1.2F1

If A→B is an isomorphism, its inverse ring map induces the inverse of u.

2.1step 1.1step 1.2∎

These two implications prove the claim.

Depends on

Used by

Dependency tree · two levels

6 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