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.

Spec A with its structure sheaf is a locally ringed space

Statement

$(Spec A, O_{Spec A})$ is a locally ringed space.

Facts & Assumptions

Given: The affine structure sheaf.

Proof

technique · direct

1.1given

At $p$ , the stalk is canonically isomorphic to [given] $A_{p}$ .

2.1step 1.1

The ring $A_{p}$ is local, with maximal ideal [step 1.1] $p A_{p}$ , so the canonically isomorphic stalk is local as well.

3.1step 2.1∎

Thus every stalk is local, which is exactly the locally ringed condition.

Depends on

Used by

Affine schemes and their coordinate rings Definition
The residue field at a point of an affine scheme Definition

Dependency tree · two levels

12 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

The Stacks Project, Definition 26.5.3 (standard reference, not scraped)