TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Morphisms to an affine scheme and global sections

Statement

For a scheme $X$ and a ring $A$ , taking global sections induces a natural bijection $Hom (X, Spec A) ≅ {Hom}_{C R i n g} (A, Γ (X, O_{X})) .$

Facts & Assumptions

Given: A scheme $X$ and a commutative ring $A$ .

Proof

technique · direct

1.1given

A morphism $f : X \to Spec A$ induces the ring map on global sections $A = Γ (Spec A, O) \to Γ (X, O_{X})$ .

2.1step 1.1construct

Conversely, a ring map $A \to Γ (X, O_{X})$ gives the canonical morphism from $X$ to the affine scheme $Spec A$ , checked on affine opens by the affine anti-equivalence and glued over an affine cover of $X$ .

3.1step 2.1∎

On each affine open the two constructions are inverse by the affine anti-equivalence; equality of scheme morphisms is local on the source, hence they are mutually inverse and natural.

Depends on

Dependency tree · 2 levels

Sources

Ravi Vakil, Foundations of Algebraic Geometry, Section 7.3.F