TheoremStatement: 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.

Sections and restrictions on distinguished opens of an affine scheme

For $f \in A$ , $Γ (D (f), O) = A_{f}$ . If $D (g) \subseteq D (f)$ , the restriction is the canonical localization map $A_{f} \to A_{g}$ .

Given: The sheaf extending the localization basis assignment.

technique · direct

1.1given

The extended sheaf agrees with the basis assignment on $D (f)$ , so its sections are $A_{f}$ .

2.1step 1.1

Its restriction along $D (g) \subseteq D (f)$ is the prescribed canonical localization map $A_{f} \to A_{g}$ .

3.1step 1.1∎

If $D (f) = \emptyset$ , then $A_{f} = 0$ , the unique ring of sections on the empty open.

Depends on