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

An affine-scheme morphism $Spec B \to Spec A$ is an isomorphism if and only if its associated homomorphism $A \to B$ is an isomorphism.

Given: A morphism $u : Spec B \to Spec A$ .

[F1]

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

technique · direct

1.1F1

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

1.2F1

If $A \to B$ 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